Cohomology Vanishing and Exceptional Sequences

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is generated by a collection of sheaves, which can either be OPn (−n),...,OPn (−1), ... morphic to a complex consisting of direct sums OPn (−i) for 0 ≤ i ≤ n (or, ...
Cohomology Vanishing and Exceptional Sequences Markus Perling

Habilitationsschrift

Fakult¨at f¨ ur Mathematik der Ruhr-Universit¨ at Bochum

Januar 2009

Contents Chapter 1. Introduction 1.1. Objective 1.2. Overview

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Chapter 2. A counterexample to King’s conjecture 2.1. Introduction 2.2. The setup 2.3. Classification of line bundles without higher cohomology 2.4. Table of cohomology-free line bundles and theorem

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Chapter 3. Divisorial cohomology vanishing on toric varieties 3.1. Introduction 3.2. Cohomology of Divisorial Sheaves 3.3. Toric 1-Circuit Varieties 3.4. Discriminants and combinatorial aspects cohomology vanishing 3.5. Arithmetic aspects of cohomology vanishing

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Chapter 4. Exceptional sequences of invertible sheaves on rational surfaces 4.1. Introduction 4.2. The birational geometry of toric surfaces 4.3. Rational surfaces and toric systems 4.4. Left-orthogonal divisors on rational surfaces 4.5. Exceptional sequences of invertible sheaves on rational surfaces 4.6. Proof of Theorem 4.5.11 4.7. Divisorial cohomology vanishing on toric surfaces 4.8. Strongly exceptional sequences of invertible sheaves on toric surfaces 4.9. Straightening of strongly left-orthogonal toric divisors 4.10. Proof of Theorem 4.8.1

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Bibliography

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CHAPTER 1

Introduction 1.1. Objective This thesis represents a snapshot of an ongoing project aimed at understanding the bounded derived category of algebraic varieties. Let X be a smooth complete algebraic variety, wich is defined over any algebraically closed field K. We denote Db (X) the bounded derived category of coherent sheaves on X. Derived categories have originally been introduced by Grothendieck and Verdier [Ver77], [Ver96] as a framework powerful enough to support the formulation of Grothendieck’s duality theory [Har66]. It turns out that the derived category, rather than being just a framework, carries a rich and interesting structure by itself. It is deeply tied to the geometric structure of X and it allows to construct links between algebraic geometry and other mathematical fields. In this work we are interested in the problem of determining certain generating sets for D b (X) for certain types of varieties. The classical example for such generating sets is given by the theorem of Be˘ılinson [Be˘ı78]. This theorem states that on Pn , the derived category is generated by a collection of sheaves, which can either be OPn (−n), . . . , OPn (−1), OPn or OPn , Ω1Pn (1), . . . , ΩbPn (n). More precisely, Be˘ılinson proves that every object in Db (Pn ) is isomorphic to a complex consisting of direct sums OPn (−i) for 0 ≤ i ≤ n (or, alternatively, of direct sums of the ΩiPn (i)). This was a striking result, which had many applications to projective geometry, in particular to the study of vector bundles and sheaves on Pn (see [OSS80]). Consequences derived from Be˘ılinson’s theorem (and the related BGG-theorem [BGG78]) remain among the most important tools in this field. One would like to have results similar to Be˘ılinson’s for other varieties too. For this, generating sets for D b (X) have been conceptualized in the form of exceptional sequences (or exceptional collections). The following definition is pivotal for the rest of this work and a large part of the cited literature: Definition: A coherent sheaf E on X is called exceptional if HomOX (E, E) = K and ExtiOX (E, E) = 0 for every i 6= 0. A sequence E1 , . . . , En of exceptional sheaves is called an exceptional sequence if ExtkOX (Ei , Ej ) = 0 for all k and for all i > j. If an exceptional sequence generates D b (X), then it is called full. A strongly exceptional sequence is an exceptional sequence such that ExtkOX (Ei , Ej ) = 0 for all k > 0 and all i, j. Note that a sequence E1 , . . . , En generates D b (X) if the smallest triangulated subcategory of containing the Ei is Db (X) itself. Of particular interest are the full strongly exceptional sequences. If E1 , . . . , En is such a sequence, then it follows by theorems of Baer [Bae88] and Bondal [Bon90] that there exists an equivalence of categories D b (X)

Ln

RHom(T , . ) : D b (X) −→ D b (A − mod),

where T := i=1 Ei and A := End(T ). This generalizes the tilting correspondence known from algebra [Hap88], [HHK07] and the sheaf T is sometimes called a tilting sheaf. In a sense (in the derived sense), the algebra A represents a non-commutative coordinization of X. 1

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CHAPTER 1. INTRODUCTION

It is of certain interest to have an explicit look on the algebras A which arise from tilting sheaves. These algebras are always finite-dimensional basic K-algebras and they can be described by quivers, i.e. by directed graphs. Given such a quiver Q with vertex set Q0 and arrows Q1 , then we can give the vector space which is spanned over all paths in the quiver the structure of an algebra structure induced by ( the path ab if h(b) = t(a) a · b := 0 otherwise. for every two arrows a, b ∈ Q1 and h, t : Q1 −→ Q0 mapping each arrow to its head and tail, respectively. Such an algebra is called a path algebra. The algebra A then is given by such an algebra modulo certain relations, generated by linear combinations paths of length at least 2. A classical example for this is the path algebra associated to the full strongly exceptional sequence OP2 , OP2 (1), OP2 (2) on P2 , which is given by: a1 a2 a3



88

&&//



b1 b2 b3

&&// 88

relations: bi aj = bj ai for i 6= j.



Another class of examples comes from the Hirzebruch surfaces. Given such a surface Fa and denote f the class of a fiber of the ruling in Pic(Fa ) and denote b the class of the base of the ruling. Then this is a strongly exceptional sequence on Fa : OFa , OFa (f ), OFa (b + (a + s)f ), OFa (b + f (a + s + 1) for any s ≥ 1. The quiver with relations is:

b0 a1 ◦ a2

11--

bs ◦

c0

.. . .. .

%%-9911

d1 ◦

11--



d2

cs+a

bi a1 = bi+1 a2 for 0 ≤ i < s ci a1 = ci+1 a2 for 0 ≤ i < s + a

d1 ci = d2 ci+1 for 0 ≤ i < s + a d1 bi = d2 bi+1 for 0 ≤ i < s

Evidently, it is easy to describe deformations of the algebra A, e.g. by multiplying factors to the terms in the relations. Then, via Db (A − mod), this can be considered as noncommutative deformations of X itself. However, the full potential of this transition to noncommutative geometry has yet to be fully explored, see for example [AKO08], [AKO06]. Figure 1.1 shows the quiver of some random strongly exceptional sequence on Sato’s toric 4-Fano [Sat00], which was found with help of a computer [Per04b]. Another motivation for the study of derived categories in algebraic geometry comes from Kontsevich’s Homological Mirror Symmetry conjecture [Kon95], see also [Kon98]. This conjecture seeks to establish a deep relation between algebraic and symplectic geometry. Roughly, the conjecture states that for an algebraic variety X there should exist another object Y which carries a symplectic structure, such that Db (X) is equivalent to the so-called derived Fukayacategory of Y . What Y precisely is, depends on X (e.g. X and Y can be a pair of Calabi-Yau varieties, or X is Fano and Y a Landau-Ginzburg model). The Homological Mirror Symmetry conjecture was motivated from physical models of superstring theory. Since its first formulation, it has found back its way into the physics literature [Dou01] and string theory papers have

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1.1. OBJECTIVE

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Figure 1.1. A computer generated quiver. become a source for examples of exceptional sequences ([HK06], [Asp08], [Asp08], [AM04], [HHV06]). The construction of exceptional sequences is notoriously difficult and only known in few cases, including certain types of homogeneous spaces [Kap85], [Kap86], [Kap88], [Kuz05], [Sam07], del Pezzo surfaces and almost del Pezzo surfaces [Gor89], [KO95], [Kul97], [KN98], and some higher dimensional Fano varieties [Nog94], [Sam05]. In general, exceptional sequences must not exist. For instance, it follows from Serre duality that there are no exceptional sheaves on a complete variety with trivial canonical bundle. Moreover, if an exceptional sequence exist, it follows that its Grothendieck group is free and finitely generated of rank n.

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CHAPTER 1. INTRODUCTION

A large class of varieties which fulfills this condition is given by toric varieties. Although exceptional sequences had been around for more than twenty years (and Be˘ılinson’s theorem for almost thirty years), it has only recently been shown by Kawamata [Kaw06] that every toric variety admits a full exceptional sequence. However, Kawamata’s work leaves (at least) two issues: 1. Kawamata’s algorithm uses the toric MMP and therefore has to utilize the larger category of toric stacks. A more direct approach for smooth toric varieties would be desirable. 2. The existence of full strongly exceptional sequences is still open. In particular, the latter point is of relevance for us, as strongly exceptional sequences provide a stronger description of D b (X) by tilting correspondence. An optional issue is that one also would like the sheaves in an exceptional collections to be as nice as possible, e.g. locally free. Indeed, on a toric variety it is reasonable to consider exceptional sequences of invertible sheaves, as these essentially share the same combinatorial description as the toric variety itself. However, the following conjecture had been open for quite a while: Conjecture ([Kin97]): Every smooth complete toric variety has a strongly exceptional sequence of invertible sheaves. A counterexample to this conjecture will be the starting point of this thesis.

1.2. Overview This thesis consists of three chapters, each representing a different stage of development of understanding of exceptional sequences and cohomology vanishing of Divisors, starting with a counterexample to King’s conjecture and ending with an almost complete understanding of exceptional sequences of invertible sheaves on rational surfaces. King’s conjecture presented a mathematical problem with some bizarre features (see the introductory sections of chapters 2 and 3 — a priori it was not even clear whether it was more of a combinatorial or of a geometric problem (which probably is the reason why it was open for several years). It turns out that it is both, combining geometry and combinatorics in a new beautiful way. In this spirit, chapters 2 and 3 are of a quite combinatorial nature. Based on the combinatorial insights, chapter 4 will provide the synthesis with the geometric aspects, thereby extending the reach of the results well beyond toric geometry. Each of the following chapters represents a separate article, which has already been published or is submitted at the time of this writing. Therefore, each chapter can be read independently.

1.2.1. Chapter 2: A counterexamle to King’s conjecture (joint work with Lutz Hille, published in Comp. Math 142(6), pp. 1507–1521, 2006). This article is completely devoted to the proof of Theorem 2.4.1, which states that there does not exist a strongly exceptional sequence of invertible sheaves on the smooth complete toric surface which is specified by the following

1.2. OVERVIEW

5

fan: 0/

// // // // −2 // // // T?O?OTOTT −3 ??OOTOTTT ?? OO TTTT ?? OOO TTTT OOO TTT ? TT

−2 −2 −1 1 The numbers indicate the self-intersection numbers of the torus invariant prime divisors corresponding to the rays in the fan. Candidates for a counterexample had been found in advance using numerical methods [Per04b]. However, the search space for exceptional sequences in Pic(X) is infinite in general and therefore cannot be exhausted by computer search. The proof is done by a brute force computation, fully exploiting the combinatorial features of toric surfaces. 1.2.2. Chapter 3: Divisorial cohomology vanishing on toric varieties (submitted). The computations of Chapter 2 had been driven by some rough, not thorough, understanding of cohomology vanishing on toric surfaces. The motivation of chapter 3 therefore is twofold: 1. To investigate phenomena encountered in the computations of Chapter 2, which are arithmetic conditions on cohomology vanishing and cohomology vanishing along certain cones in the inverse nef cone. 2. To provide some better foundations for numerical investigations and to better understand their possible shortcomings. Let X be a d-dimensional toric variety on which the algebraic torus T acts and denote D a torus invariant Weil divisor. The basic idea is that the action of T induces an isotypical decomposition on the cohomologies of the sheaf O(D): M   H i X, O(D) ∼ H i X, O(D) m , = m∈M

where M ∼ = Zd denotes the character group of T . Recall that to X there is associated a fan ∆, whose cones are spanned by primitive vectors l1 , . . . , ln , which correspond one-to-one to torus invariant prime divisors Pn D1 , . . . , Dn on X. Every divisor D ∈ Ad−1 (X) has a torus invariant representative D = i=1 ci Di . Then for every m ∈ M there is associated a signature I = I(m) ⊂ {1, . . . , n}, given by I = {i | li (m) < −ci }. This way, we can associate to every character  m ∈ M associate a unique subcomplex ∆I of ∆ such that H i X, O(D) m ∼ = H i−1 (∆I ; K) for every m ∈ M (see Theorem 3.2.11). This shows that the cohomology vanishing problem is governed by the hyperplane arrangement in M ⊗Z Q, which is given by the hyperplanes c

Hi := {m ∈ MQ | li (m) = −ci }. It turns out that the combinatorial types of these arrangements induce a stratification of the Q-divisors class group via the short exact sequence: 0 −→ M ⊗Z Q −→ Qn −→ Ad−1 (X) ⊗Z Q −→ 0, where the inclusion is given by the tupel (l1 , . . . , ln ). The relevant stratification is given by the so-called discriminantal arrangement in Ad−1 (X) ⊗Z Q which, coined by Manin and Schechtman [MS89]. The hyperplanes in this arrangement are in one-to-one correspondence with the minimal linearly dependent subsets of {l1 , . . . , ln }, so-called circuits. Essentially all properties

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CHAPTER 1. INTRODUCTION

of divisors are governed by this stratification, in particular, one identifies the effective cone, the secondary fan, and the nef cone as its substructures. Another ingredient are arithmetic conditions coming from the so-called diophantine Frobenius problem. For us, roughly, the diophantine Frobenius problem is to determine whether a c given chamber of the arrangement given by the Hi contains lattice points of M or not. We even c consider a simplified problem, where we ask this questions for subarrangements given by Hi for i ∈ C, where C ⊂ {1, . . . , n} such that the set {li }i∈C forms a circuit. This allows to define arithmetic thickenings in Ad−1 (X) of discriminantal strata in Ad−1 (X) ⊗Z Q. In particular, for every discriminantal stratum S we can define its arithmetic core, AS in Ad−1 (X) which is the intersection of all thickened half spaces containing S. This way we obtain the following arithmetic version of the toric Kawamata-Viehweg vanishing theorem:  Theorem (3.5.11): Let X be a complete toric variety and D ∈ Anef . Then H i X, OX (D) = 0 for all i > 0. This version of the Kawamata-Viehweg vanishing gives somewhat refined conditions as compared to the usual toric Kawamata-Viehweg theorem. The following theorem is new, as it makes a statement of vanishing related to the inverse nef cone:  Theorem (3.5.14): Let X be a d-dimensional toric variety. Then H i X, O(D) = 0 for every i and all D which are contained in some A−F , where F is a face of nef(X) which consists cointains divisors of Iitaka dimension 0 < κ(D) < d. If A−F is nonempty, then it contains infinitely many divisor classes. These two theorems capture almost all divisors on toric surfaces which have vanishing higher cohomology classes: Theorem (3.5.18): Let X be a complete toric surface. Then there are only finitely S many divisors D with H i X, OX (D) = 0 for all i > 0 which are not contained in Anef ∪ F A−F , where the union ranges over all faces of nef(X) which correspond to pairs of opposite rays in the fan associated to X. This conceptualizes and generalizes the appearance of infinite families of invertible sheaves without higher cohomologies in the classification of chapter 2. The final main result of chaper 3 is the following statement about maximally Cohen-Macaulay modules. Theorem (3.5.31): Let X be a d-dimensional affine toric variety whose associated cone has simplicial facets and let D ∈ Ad−1 (X). If Ri π∗ OX˜ (π ∗ D) = 0 for every regular triangulation ˜ −→ X, then OX (D) is MCM. For d = 3 the converse is also true. π:X The rationale here is that our approach treats global and local cohomology vanishing on equal footing. 1.2.3. Chapter 4: Exceptional sequences of invertible sheaves on rational surfaces(joint work with Lutz Hille, submitted). Chapter 4 contains the main results of this work. The question we are going to answer is whether on rational surfaces there exist exceptional and strongly exceptional sequences which consist of invertible sheaves. The basic lesson learned from chapter 3 is that, at least for the toric case, Gale duality should play a role. This can be achieved as follows. Let X be a smooth complete rational surface (which is not necessarily toric), and denote E1 , . . . , En ∈ Pic(X) a set of divisors such that O(E1 ), . . . , O(En ) forms a full exceptional sequence. To bring this sequence into a normal form, we do the following trick:

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we set Ai := Ei+1 − Ei for 1 ≤ i < n and An := −KX − of divisors A1 , . . . , An has the following properties:

Pn−1 i=1

Ai . It turns out that this system

(i) Ai .Ai+1 = 1 for 1 ≤ i < n and A1 .An = 1; (ii) P Ai .Aj = 0 for i 6= j, {i, j} = 6 {1, n}, and {i, j} = 6 {k, k + 1} for any 1 ≤ k < n; n (iii) A = −K . i X i=1

Definition: We call a set of divisors on X which satisfy the conditions (i), (ii), (iii) above a toric system. Now we apply Gale duality via the following short exact sequence: 0 −→ Pic(X) −→ Zn −→ Z2 −→ 0 where the inclusion is given by D 7→ (D.A1 , . . . , D.An ). We will show that the images l1 , . . . , ln of the standard basis of Zn in Z2 generate the fan of a smooth complete toric surface. We get even more: Theorem (4.3.5): Let X be a smooth complete rational surface, let OX (E1 ), . . . , OX (En ) be a full exceptional sequence of invertible sheaves on X, and set En+1 := E1 − KX . Then to this sequence there is associated in a canonical way a smooth complete toric  surface with torus 2 invariant prime divisors D1 , . . . , Dn such that Di + 2 = χ OX (Ei+1 − Ei ) for all 1 ≤ i ≤ n.

This result is very surprising and, despite the simplicity of the construction, it seems not to have been noticed before. It is very intriguing to ponder whether there might be a new universal property of toric varieties behind this result. For now, it allows us to derive various criteria for the existence of exceptional sequence of invertible sheaves:

Theorem (4.5.6): On every smooth complete rational surface exists a full exceptional sequence of invertible sheaves. Theorem (4.5.9): Any smooth complete rational surface which can be obtained by blowing up a Hirzebruch surface two times (in possibly several points in each step) has a full strongly exceptional sequence of invertible sheaves. In the toric case, we can show that the converse is also true: Theorem (4.8.2): Let P2 6= X be a smooth complete toric surface. Then there exists a full strongly exceptional sequence of invertible sheaves on X if and only if X can be obtained from a Hirzebruch surface in at most two steps by blowing up torus fixed points. Another type of results are related to what we call cyclic exceptional sequences. These are in particular relevant for applications coming from superstring theory, as mentioned above: Definition: An infinite sequence of sheaves . . . , Ei , Ei+1 , . . . is called a cyclic (strongly) exceptional sequence if Ei+n ∼ = Ei ⊗O(−KX ) for every i ∈ Z and if every winding (i.e. every subinterval Ei+1 , . . . , Ei+n ) forms a (strongly) exceptional sequence. A cyclic exceptional sequence is full if every winding is a full exceptional sequence. Here, we also obtain various critera: Theorem (4.5.13): Let X be a smooth complete rational surface on which a full cyclic strongly exceptional sequence of invertible sheaves exists. Then rk Pic(X) ≤ 7. So not even every del Pezzo surface admits such a sequence. However:

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CHAPTER 1. INTRODUCTION

Theorem (4.5.14): Let X be a del Pezzos surface with rk Pic(X) ≤ 7, then there exists a full cyclic strongly exceptional sequence of invertible sheaves on X. The condition that −KX is ample can be weakened in general. In the toric case we obtain a complete characterization for toric surfaces admitting cyclic strongly exceptional sequences: Theorem (4.8.5 & 4.8.6): Let X be a smooth complete toric surface, then there exists a full cyclic strongly exceptional sequence of invertible sheaves on X if and only if −KX is nef. For further explanations and examples we refer the reader to chapter 4.

CHAPTER 2

A counterexample to King’s conjecture 2.1. Introduction It is a widely open question whether on a given smooth algebraic variety X (say, complete and smooth), there exists a tilting sheaf. A tilting sheaf is a sheaf T which generates the bounded derived category D b (X) of X and Extk (T , T ) = 0 for all k > 0. For such T , the functor RHom(T , . ) : D b (X) −→ D b (A − mod), where A := End(T ) is the endomorphism algebra, induces an equivalence of categories (see [Rud90], [Bon90], [Be˘ı78]). The existence of a tilting sheaf implies that the Grothendieck group of X is finitely generated and free, so that in general such sheaves cannot exist. However, so far there are a number of positive examples known, including projective spaces, del Pezzos, certain homogeneous spaces, and some higher dimensional Fanos. An obvious testbed for the existence of tilting sheaves are the toric varieties. There is a quite strong conjecture which was first stated by King [Kin97]: Conjecture: Let X be a smooth complete toric variety. Then X has a tilting sheaf which is a direct sum of line bundles. its direct summands T = Lt If a tilting sheaf decomposes into a direct sum of line bundles, k L form a so-called strongly exceptional sequence, i.e. Ext (L , Lj ) = 0 for all i, j and all i i i=1 k > 0, and — after possibly reordering the Li — Hom(Li , Lj ) = 0 for i > j. Moreover, t is the rank of the Grothendieck group of X. It would be very nice if there existed easy-computable tilting sheaves on toric varieties, and indeed there are known a lot of positive examples in favor of the conjecture (see [CM04], [Kaw06], [Hil04], [CS05], and also [AH99], [AKO08] and [BP08] for related results). Computer experiments also look promising in many directions. However, the conjecture remained somewhat mysterious so far and, as it turns out, it is false in general. It is the purpose of this paper to present a counterexample. Our counterexample is the toric surface X as shown in figure 2.1, which can be obtained by iteratively blowing up the Hirzebruch surface F2 three times. In coordinates, the primitive vectors of its rays are given as shown in figure 2.1. Note that the rank of the Grothendieck group of X is 7. To show that there do not exist any strongly exceptional sequences of length 7 on this surface, we will perform explicit computations in the Picard group to determine cohomology vanishing. More precisely, note that if L1 , . . . , Lt is a strongly exceptional sequence, then also L1 ⊗ L′ , . . . , Lt ⊗ L′ is strongly exact, where L′ is any line bundle. So one can assume without loss of generality that the sequence contains the structure sheaf. Then a necessary condition for the bundles in the sequence is that all the higher cohomology groups of the bundles and of their dual bundles vanish, i.e. H k (X, Li ) = H k (X, L∗i ) = 0 for all i and all k > 0. This is a rather strong condition, and our main computation will be to compile a complete list of such bundles for our surface X. After 9

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CHAPTER 2. A COUNTEREXAMPLE TO KING’S CONJECTURE 6

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Figure 2.1. The fan. The endpoints of the lines indicate the position of the primitive vectors: l1 = (1, −1), l2 = (2, −1), l3 = (3, −1), l4 = (1, 0), l5 = (0, 1), l6 = (−1, 2), l7 = (0, −1) having obtained this classification, we deduce by simple inspection that a strongly exceptional sequence of length 7 and consisting of line bundles does not exist. Overview: In section 2.2 we state everything we need to know about cohomology of line bundles on toric surfaces and we describe in more detail our method of computation. In section 2.3 all bundles are classified which have the property that the higher cohomologies of the bundles themselves and of their dual bundles vanish. In section 2.4 we present the complete classification obtained in section 2.3 and we show by inspection that there exist no strongly exceptional sequences of length 7 on X. 2.2. The setup In this section we recall basic facts on cohomology of line bundles on a toric surface and we describe our method of computation. For general information about toric varieties we refer to the books [Oda88], [Ful93]. 2.2.1. Generalities on toric line bundles. Let X be a complete smooth toric surface on which the torus T acts. The variety X is described by a fan ∆ which is contained in a 2dimensional vector space NR := N ⊗Z R, where N ∼ = Z2 is the group of 1-parameter subgroups of T . We denote by ∆(1) the set of rays, that is, of one-dimensional cones of ∆. As X is a complete surface, the fan is completely determined by the rays. We denote the primitive vectors of the rays by l1 , . . . , ln , enumerated in counterclockwise order. To any li there is associated a T -invariant divisor Di , and every divisor D can, up to linear equivalence, written as a sum of Pn these invariant divisors, i.e. D = i=1 ci Di . We denote M ∼ = Z2 the character group of the torus acting on X and we set MR := M ⊗Z R. The lattice N is in a natural way dual to M , and the primitive vectors li are integral linear forms on M (and on MR , respectively). There is a short exact sequence A

0 −→ M −→ Z∆(1) −→ Pic(X) −→ 0,

2.2. THE SETUP

11

where the matrix A is composed of the li as row vectors. This sequence is split exact. More precisely, if we choose two of the li , for instance ln−1 and ln , which form a Z-basis of N , then the divisors D1 , . . . , Dn−2 form a Z-basis of Pic(X). So every divisor D has a unique representation Pn−2 D = i=1 ci Di . P Now let D = ni=1 ci Di be any T -invariant divisor. D in a natural way defines an affine hyperplane arrangement HD = {H1 , . . . , Hn } in the vector space MR , where Hi = {m ∈ MR | li (m) = −ci }. All information on the cohomology of the line bundle O(D) is contained in the chamber structure HD (or more precisely, in the intersection of this chamber structure with the lattice M ). Recall that the T -action induces an eigenspace decomposition on the cohomology groups of O(D): M   H k X, O(D) = H k X, O(D) m m∈M

 for all k ≥ 0. The dimension of H k X, O(D) m as a vector space is determined by the signature of m with respect to the arrangement HD : Pn Definition 2.2.1: Let D = i=1 ci Di be a T -invariant divisor on X. Then for every i = 1, . . . , n we define a signature ΣD i : M −→ {+, −, 0},

D D where ΣD i (m) = + if li (m) > −ci , Σi (m) = − if li (m) < −ci and Σi (m) = 0 if li (m) = −ci . Moreover, we denote

ΣD : M −→ {+, −, 0}n ,  D where ΣD (m) is the tuple ΣD 1 (m), . . . , Σn (m) .

Below we will mostly work with only one D at a time, whose values for the ci will be clear from the context. So usually we will omit the reference to D in the notation, i.e. we will mostly write Σ(m) instead of ΣD (m).  Given the signature ΣD (m), the computation of H k X, O(D) m is straightforward. We use a method which for toric surfaces can easily be derived from the standard textbook treatment for cohomology computation for line bundles (we refer to [EMS00] and [HKP05] for an explicit description for general toric varieties, see also [AH99], lemma 3.4). For a given signature ΣD (m), a −-interval is a connected sequence of − with respect to the circular order of the ρi . For example, assume that ∆(1) consists of 7 elements enumerated in circular order. Then the signature + − − + + − + has two −-intervals. Note that due to the circular ordering of the rays, the signature − − + + + − − has only one −-interval. Lemma 2.2.2: Let D be a T -invariant divisor on a toric surface X and m ∈ M . Then we have: (  the number of −-intervals − 1 if there exists at least one −-interval, 1 dim H X, O(D) m = 0 else. and

 dim H X, O(D) m = 2

( 1 0

if ΣD (m) = {−}n else.

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CHAPTER 2. A COUNTEREXAMPLE TO KING’S CONJECTURE

2.2.2. Method of computation. Let L1 , . . . , Lt be a strongly exceptional sequence of line bundles, i.e. we have Extk (Li , Lj ) = 0 for all i, j and all k > 0. There is a natural isomorphism Extk (Li , Lj ) ∼ = H k (X, L∗i ⊗ Lj ), where L∗i = Hom(Li , OX ) denotes the dual bundle. By this we can assume without loss of generality that one of the Li is just the structure sheaf OX , i.e. L1 , . . . , Lt is a strongly exceptional sequence if and only if L∗i ⊗ L1 , . . . , L∗i ⊗ Lt is a strongly exceptional sequence. If OX is part of the sequence, this in turn implies a rather strong condition on the cohomologies of the other bundles. Namely, for every Li we have: H k (X, Li ) = H k (X, L∗i ) = 0 for all k > 0. Thus, to show that our toric surface does not have a strongly exceptional sequence of length 7, we proceed in 2 steps: (i) We classify all line bundles where higher cohomologies of the bundle itself as well as of its dual vanish. It turns out that the list of such bundles has a rather short description, although it is not finite. (ii) After having obtained the list, we show by exclusion that there are no strongly exceptional sequences of length 7. H4

H3--++++-+++++-

---+++-

++++++-

----+++++++--

H5/H7 -----++ ++++--+

H6 ------+

+++---+

H1

+-----+

H2

++----+

Figure 2.2. The central arrangement Example: Figure 2.2 shows the arrangement which belongs to the structure sheaf. We see that this arrangement is central and induces a chamber decomposition of the space MR , consisting of unbounded chambers. To every chamber there is associated a signature which we have indicated in the picture. Note that in fact there are some more signatures which are not shown. For instance, the points lying on the line between the chambers with signatures + + + + + + − and − + + + + + − have signature 0 + + + + + −. The origin has signature 0000000. Figure 2.3 shows a deformation of this central arrangement which belongs to the divisor D = −(4D1 + 7D2 + 11D3 + 4D4 + 2D5 ). As we can see, moving the hyperplanes creates new chambers with new signatures. There are two new unbounded chambers with signatures

13

2.2. THE SETUP

H4

H3

00++++-

H2

H1

++++++0

H6 +++++00

0--0++0

H7 +000+++

+++++++

----+++

+++++0+

++++-+

H5 +++000+

Figure 2.3. A deformation of the central arrangement − − − − + + + and + + + + + − +, respectively, which obviously have no influence on the comohology of O(D). The other chambers are all bounded and thus contain only a finite number of lattice points (i.e. points in M ). We have indicated the signatures of some of these points in the picture. As one can check, most of these signatures give not rise to nonvanishing cohomology, the only exception being the point with signature + + + + + + +. Recall that we are interested in the classification of line bundles which have no higher cohomology and whose duals have also no higher cohomology. So, if there is an inequality li (m) < −ci (or li (m) > −ci , respectively), then we have li (−m) > ci (li (−m) < ci , respectively), whereas for li (m) = −ci we have li (−m) = ci . In our example the signature of m with ΣD (m) = + + + + + + + becomes Σ−D (−m) = − − − − − − − for the dual bundle, which therefore has nonvanishing H 2 . We give one more example and some more notation. In many situation it will not be necessary to know the complete signature of some point m ∈ M . Therefore we define: Definition 2.2.3: A partial signature is given by ΣD : M −→ {+, −, 0, ∗}n  which is a signature for some subset I of {1, . . . , n} such that ΣD (m) i = ΣD i (m) for i ∈ I and  D Σ (m) i = ∗ for i ∈ / I.

For us it is convenient to use the same symbol for signatures and partial signatures. To exemplify our computations, we prove the following Lemma 2.2.4: c5 ≤ 5.

14

CHAPTER 2. A COUNTEREXAMPLE TO KING’S CONJECTURE

P Proof. Assume D = 5i=1 ci Di and c5 > 0. Now consider the point m in M which has the coordinates (1 − 2c5 , −c5 ) (see figure 2.4). H6 H7

(0,0) (-3,-1)

H5

(1-2c_5, -c_5)

Figure 2.4. A partial arrangement Its partial signature with respect to the linear forms l5 , l6 , l7 is Σ(m) = ∗ ∗ ∗ ∗ 0 − +. Our aim is to derive conditions on the values of the ci . Evidently, any complete signature which is obtained by filling the ∗’s has at least one −-interval. Moreover, if any of the ∗’s becomes a −, the signature has at least two −-intervals, and any corresponding line bundle will have nonvanishing H 1 . So, a necessary condition is that ΣD i (m) ∈ {+, 0} for i = 1, . . . , 4 and any valid divisor D. This in turn implies: c1 ≥ c5 − 1 c2 ≥ 3c5 − 2 (1) c3 ≥ 5c5 − 3 c4 ≥ 2c5 − 1. Now the point (−3, −1) has partial signature Σ(−3, −1) = ∗∗∗∗+++, and the above conditions on c1 , . . . , c4 imply that for c5 > 3 this point always has signature + + + + + + +, and thus we have nonvanishing H 2 . Hence, we conclude c5 ≤ 3.  2.3. Classification of line bundles without higher cohomology In this section we do the complete classification of line bundles for our toric surface which have the property that the higher cohomologies vanish for both, the bundle itself and its dual. The result is listed in table 1 at the end of this paper. As explained in the previous section, we can P always assume that a line bundle L is uniquely represented by an invariant divisor D = 5i=1 ci Di , and every tuple of numbers (c1 , . . . , c5 ) represents a unique isomorphism class in Pic(X). As we already have seen in lemma 2.2.4, we can assume that c5 ≤ 3. Moreover, as it does not matter if we deal with a bundle or its dual, we can without loss of generality assume c5 ≥ 0. So, this leaves us with four possible values for c5 . Our classification will be done by subsequential case distinctions which on the toplevel are guided by the four possible values of c5 . Note that in the sequel for a given bundle we will use phrases like “has cohomology” if either the bundle itself or its dual has a nonvanishing higher cohomology group. We prove the following theorem:

2.3. CLASSIFICATION OF LINE BUNDLES WITHOUT HIGHER COHOMOLOGY

15

Theorem 2.3.1: Table 1 contains all line bundles on X with no higher cohomology groups. The rest of this section is devoted to the proof of theorem 2.3.1. 2.3.1. c5 = 3. Recall that the partial signature of the point (−3, −1) is ΣD (−3, −1) = ∗ ∗ ∗ ∗ + + +. By the conditions (1), we immediately obtain the partial signature ∗ + + + + + +. So, the only way to prevent H 2 to show up in the dual bundle is ΣD 1 (−3, −1) = 0 (then, for the −D dual bundle, we have the signature Σ (3, 1) = 0 − − − − − −). This in turn means that c1 = 2. But then, we have ΣD (−4, −2) = 0 + + + +0+, and thus Σ−D (4, 2) = 0 − − − −0−, hence we get H 1 . We conclude that there are no divisors with c5 = 3 and vanishing cohomology. 2.3.2. c5 = 2. Here, conditions (1) read: c1 c2 c3 c4

≥ 1, ≥ 4, ≥ 7, ≥ 3.

We first consider c4 = 3. Then Σ(−3, 0) = ∗ ∗ ∗0 + +0. If one of the ∗’s is replaced by +, this implies that the dual signature will have at least two −-intervals, independent on the other substitutions. So we obtain: c1 ≤ 3, c2 ≤ 6, c3 ≤ 9. We treat these 27 possibilities case by case. First, let c1 = 1. Then we have Σ(−2, −1) = 0 ∗ ∗ + +0+, and so we have H 1 , leaving only 18 more cases. For these we write a table: c1 c2 c3 m Σ(m) 2 4 8 (−3, −2) +0 + 00 − + 2 6 — (−3, −1) 0 + ∗0 + ++ 2 — 9 (−3, −1) 0 ∗ +0 + ++ 3 4 — (−2, 0) +0 ∗ + + +0 3 5 9 (−3, −1) +0 + 0 + ++ 3 6 7 (−2, 1) 0 + 0 + + + − This table contains a list of all values which have cohomology. For given values of c1 , c2 , c3 , the fourth columns contains a lattice point m ∈ M with bad signature, which is displayed in the fifth column. Sometimes it suffices to display only a partial signature. Then the box for the corresponding ci contains a dash (—). All tuples which are not displayed in the above table represent cohomology free line bundles, namely: c1 c2 c3 c4 c5 2 4 7 3 2 2 5 7 3 2 2 5 8 3 2 3 5 7 3 2 3 5 8 3 2 3 6 8 3 2 3 6 9 3 2

16

CHAPTER 2. A COUNTEREXAMPLE TO KING’S CONJECTURE

Now for c4 = 4. We have Σ(−3, −2) = ∗ ∗ ∗ + 0 − + and thus we get the bounds c1 ≥ 2, c2 ≥ 5, c3 ≥ 8. Moreover, we have Σ(−4, 0) = ∗ ∗ ∗0 + +0 and thus c1 ≤ 4, c2 ≤ 8, c3 ≤ 12. Further, Σ(−3, −1) = ∗ ∗ ∗ + + + + and hence c1 ≤ 2 or c2 ≤ 5 or c3 ≤ 8. We have Σ(−4, −3) = ∗ ∗ ∗0 − −+ which implies c3 ≥ 9 and so the case c3 ≤ 8 cannot occur. Also, the conditions imply that either c1 = 2 or c2 = 5, thus leaving 24 possibilities. We first consider the case c1 = 2. Then we have Σ(−4, −2) = 0 ∗ ∗000+, which implies c2 ≤ 6 and c3 ≤ 10. Now we take c2 = 5. Then Σ(−4, −3) = +0 ∗ 0 − −+, so that we must have c3 = 9. For c2 = 6, we have Σ(−3, −1) = 0 + ∗ + + + + which implies c3 ≥ 9. Indeed, we have found: c1 c2 c3 c4 c5 2 5 9 4 2 2 6 9 4 2 2 6 10 4 2 Now we consider c2 = 5. We can assume that c1 ≥ 3. Assume that c3 ≥ 10. Then Σ(−4, −3) = +0 + 0 − −+, so we have cohomology, hence c3 = 9. For c1 = 4, we have Σ(−4, −3) = +0 + 0 − −+ and thus cohomology, hence c1 = 3, and indeed we have found: c1 c2 c3 c4 c5 3 5 9 4 2 Now we go on with c4 ≥ 5. Then we have Σ(−4, −2) = ∗ ∗ ∗ + 00+, which yields the conditions c1 ≥ 4, c2 ≥ 7, c3 ≥ 11. The signature Σ(−3, −1) as before implies c1 ≤ 2 or c2 ≤ 5 or c3 ≤ 8. both conditions cannot be fulfilled simultaneously, and hence, for c4 ≥ 5 there are no cohomologyfree bundles.

2.3. CLASSIFICATION OF LINE BUNDLES WITHOUT HIGHER COHOMOLOGY

17

2.3.3. c5 = 1. Again, we start with the conditions (1), which read c1 c2 c3 c4

≥ 0, ≥ 1, ≥ 2, ≥ 1.

Now we go for the different cases for c1 . c1 = 0. Then we have Σ(−1, −1) = 0 ∗ ∗ ∗ 0 − + so that all of the ∗’s can only be substituted by 0’s, and thus c2 = 1, c3 = 2, c4 = 1 and indeed we have found: c1 c2 c3 c4 c5 0 1 2 1 1 with no other possibilities left. c1 = 1. We have Σ(−2, −1) = 0 ∗ ∗ ∗ 00+ which implies c2 ≤ 3, c3 ≤ 5, c4 ≤ 2. Let c4 = 1, then Σ(−1, 0) = 0 ∗ ∗0 + +0, which implies c2 ≤ 2, c3 ≤ 3. From these four cases, only c2 = 1, c3 = 2 has cohomology, as in this case Σ(−1, −1) = +0 + 00 − +. We have found: c1 c2 c3 c4 c5 1 1 2 1 1 1 2 2 1 1 1 2 3 1 1 Now let c4 = 2. Then Σ(−2, −1) = + ∗ ∗ + 0 − 0, so that c2 ≥ 2, c3 ≥ 3, leaving six cases. We write a table as before: c2 c3 m Σ(m) 2 3 (−3, −2) +0 − 0 − −+ 2 5 (−3, −2) +0 + 0 − −+ 3 3 (−1, 0) 0 + 0 + + + 0 and thus we have found: c1 c2 c3 c4 c5 1 2 4 2 1 1 3 4 2 1 1 3 5 2 1

18

CHAPTER 2. A COUNTEREXAMPLE TO KING’S CONJECTURE

c1 ≥ 2. Now for any c1 ≥ 2, the point (1 − c1 , 0) has signature Σ(1 − c1 , 0) = + ∗ ∗ ∗ + + 0. So, we obtain general conditions c2 ≥ 2c1 − 1, c3 ≥ 3c1 − 2, c4 ≥ c1 . We obtain another general condition as follows. Consider the signature Σ(−c1 − 1, 0) = − ∗ ∗ ∗ + + 0. Assume that c4 ≥ c1 + 2. Then Σ(−c1 , 0) = − + ∗ ∗ + + 0 and so the ∗’s can only be replaced by +’s, hence c2 ≥ 2c1 + 3. The signature Σ(−c2 − 2, −1) then becomes either −0 ∗ −0 + 0 or − + ∗ − 0 + 0 which both are bad. Thus c4 must be strictly smaller than c1 + 2, and we have: c4 ∈ {c1 , c1 + 1} for any value of c1 ≥ 2. Now consider the signature Σ(−c1 − 1, −1) = 0∗∗∗0+ +, which yields the following restrictions: c2 ≤ 2c1 + 1, c3 ≤ 3c1 + 2. Now assume that c4 = c1 . From the signature Σ(−c1 , 0) = 0 ∗ ∗0 + +0 we get immediately the conditions c2 ≤ 2c1 , c3 ≤ 3c1 . If c2 = 2c1 − 1, we have the signature Σ(−c1 , −1) = +0 ∗ 000+, respectively Σ(−2, −1) = +0 ∗ 00 + +, for the case c1 = 2. In either case, we get: c3 ≤ 3c1 − 1. For c2 = 2c1 , we have the signature Σ(1 − c1 , 1) = 0 + ∗ + + + −, hence the ∗ cannot be replaced by − or 0, thus we get c3 ≥ 2c1 − 1. We cannot find any more restrictions and in fact we have found inifinite series of cohomology-free line bundles: c1 c2 c3 c4 c5 k ≥ 2 2k − 1 3k − 2 k 1 k ≥ 2 2k − 1 3k − 1 k 1 k≥2 2k 3k − 1 k 1 k≥2 2k 3k k 1 Now let c4 = c1 + 1. The signature Σ(−c4 , −1) = + ∗ ∗0 − −+ yields c2 ≥ 2c1 , c3 ≥ 3c1 + 1. This leaves four possibilities of which we can only exclude the case c2 = 2c1 , c2 = 3c1 + 2. Here we distinguish cases c1 = 2, 3, ≥ 4. For c1 = 2, we have Σ(−3, −2) = +0 + 0 − −+, for c1 = 3 we have Σ(−4, −2) = +0 + 00 − + and for c1 ≥ 4 we have Σ(−c1 − 1, −2) = +0 + 0 + −+, all of which are bad signatures. So, we have extracted three more series: c1 c2 c3 c4 c5 k≥2 2k 3k + 1 k + 1 1 k ≥ 2 2k + 1 3k + 1 k + 1 1 k ≥ 2 2k + 1 3k + 2 k + 1 1

2.3. CLASSIFICATION OF LINE BUNDLES WITHOUT HIGHER COHOMOLOGY

19

2.3.4. c5 = 0. . We have the signatures Σ(−1, 0) = ∗ ∗ ∗ ∗ 0 + 0 and Σ(1, 0) = ∗ ∗ ∗ ∗ 0 − 0 which imply: −1 ≤ c1 −2 ≤ c2 −3 ≤ c3 −1 ≤ c4

≤ 1, ≤ 2, ≤ 3, ≤ 1.

As c5 = 0, we can assume without loss of generality c1 ≥ 0. We refine by case distinction by the values of c1 . c1 = 0. Here we can assume without loss of generality that c4 ≥ 0. Let c4 = 0 and thus without loss of generality c2 ≥ 0. We have the following table c2 c3 m Σ(m) 0 2 (−1, −1) 0 − 0 − − − + 0 3 (−1, −1) 0 − + − − − + 1 ≤0 (1, 0) −0 − 0 + +− 1 ≥2 (0, 1) −0 + 0 + +− Thus we have found: c1 c2 c3 c4 c5 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 Now let c4 = 1. Then Σ(−1, −1) = 0 ∗ ∗0 − −+ and hence c2 = 1 and c3 = 2. We have found: c1 c2 c3 c4 c5 0 1 2 1 0 c1 = 1. Assume first that c4 = −1. Then Σ(0, 1) = 0 ∗ ∗ − + + − which makes c4 = −1 impossible. Now let c4 = 0. We have Σ(0, 1) = 0 ∗ ∗0 + +− which implies that c2 = 1 and c3 = 1. We have found c1 c2 c3 c4 c5 1 1 1 0 0 Finally, let c4 = 1. Then Σ(0, 0) = + ∗ ∗ + 000, so c2 ≥ 1, c3 ≥ 1. So we have reduced to six possibilities. Consider c2 c3 m 1 1 (−1, −1) 1 3 (−1, −1) 2 1 (0, 2) The remaining cases are: c1 c2 c3 1 1 2 1 2 2 1 2 3 which finishes the classification.

the table Σ(m) +0 − 0 − −+ +0 + 0 − −+ −0 − + + +− c4 c5 1 0 1 0 1 0

20

CHAPTER 2. A COUNTEREXAMPLE TO KING’S CONJECTURE

2.4. Table of cohomology-free line bundles and theorem We represent the classification obtained in the previous section in a table at the end of this section. We distinguish three types of line bundles, named by the letters A to C, where the B-type bundles form infinite series. For a given cohomology-free bundle L the table shows the tuple (c1 , c2 , c3 , c4 , c5 ) and a list all cohomology-free bundles L′ which have the property that H k (X, L∗ ⊗ L′ ) = H k (X, L ⊗ (L′ )∗ ) = 0 for all k > 0, which is a necessary condition for L and L′ for being part of the same strongly exceptional sequence. We say that L and L′ are compatible. Note that for the presentation purposes for the B-type bundles we have only listed the compatible bundles which are also of type B. For notation, −A4 for instance means the line bundle (−1, −1, −1, 0, 0). Now we state and proof our main result. Let X be the toric surface as given in the introduction. Theorem 2.4.1: On X there are no strongly exceptional sequences of length 7 which consist of line bundles. Proof. The proof is done by inspection of the table and exclusion principle. For example, assume that we have a strongly exceptional sequence of length 7 which contains C11 . Then the rest of the sequence can at most be selected from A1 , A2 , A4 , C3 , C8 , C10 , B4,1 , B4,2 . We see from the corresponding rows that at most one of the Ai and at most one of the Ci can be selected simultaneously. Hence we can chose at most four elements from the list to complete the sequence. We conclude that a strongly exceptional sequence of length 7 which contains C11 cannot exist. Thus we can eliminate C11 from the table. As general rules we read off that at most two of the Ai can be part of a strongly exceptional sequence, i.e. we have either ±Ai , i = 1, . . . , 7 alone or Ai , i = 1, . . . , 7, and A7 (respectively −Ai and −A7 ), together, or Ai and −A7−i , i = 1, . . . , 6 (respectively −Ai and A7−i ), together. Assume that a strongly exceptional sequence contains three bundles of type Br,k , Bs,l , Bt,m . We read immediately off from the table that this is not possible if r, s, t are pairwise distinct, hence at least two of the r, s, t coincide. We also see that always Br,k+1 − Br,k = A7 for all r and Br,k+n − Br,k = n · A7 , so if two bundles of the same B-type are contained in a strongly exceptional sequence, these must be of the form Br,k , Br,k+1 . Now given such a pair and assume that there exists one more Bs,l together with this pair in a strongly exceptional sequence. Then Br,k+1 − Bs,l = Ai for some 1 ≤ i ≤ 6 and Br,k − Bs,l = −A7−i . If there exists another Bt,m in this sequence, we have Br,k+1 − Bt,m = Aj for 1 ≤ j ≤ 6 and Bt,m − Bs,l = Ai − Aj , which is not possible. So we conclude that a strongly exceptional sequence can contain at most three of the B’s. This in turn, together with the above condition on the A’s, implies that a full strongly exceptional sequence must contain at least one of the C’s. We proceed now with C10 . We can choose at most three of the compatible B’s and at most one of the A’s. So we have to choose at least one out of C2 and C7 . Again, these two are mutally exclusive, so we can choose only one of them. Both choices restrict the choice of the A’s to −A1 , which in turn is not compatible with B4,k . So that we can choose at most two of the B’s, which is not enough, hence we can forget about C10 . For C9 , we can choose at mosts three of the B’s and thus to obtain a strongly exceptional sequence, we have to choose both, A5 and C1 . But A5 is not compatible with B2 , so we cannot complete to a full sequence. Hence we eliminate C9 . C8 . The bundles C1 and C7 are mutually exclusive, so in order to obtain an exceptional sequence of length seven, we have to choose one out of the A’s and three out of the B’s. The C’s leave only one choice for the A’s, namely −A2 , which in turn is not compatible with B4,k , hence we can discard C8 .

2.4. TABLE OF COHOMOLOGY-FREE LINE BUNDLES AND THEOREM

21

C7 . Here we have only the choice of at most one of the A’s and of at most three of the B’s left, which is not enough. So C7 goes away. C6 . We can choose at most three of the compatible B’s and at most one of the A’s. So we have to choose at least one out of C2 and C5 , which mutually exclude each other. So we can exclude C6 . C5 . We have just excluded C6 . Both pairs C3 , C4 and B1,2 , B7,2 are mutually exclusive, leaving not enough choices to complete the sequences. Bye bye, C5 . C4 . The sequence must contain C1 and −A2 , where the latter is not compatible with the B7 ’s, so no C4 . C3 . We can choose at most one A and at most one C. The C’s are not compatible with A1 and A2 , and B4 and B7 are not simultaneously compatible with one of −A3 and −A4 , which does not leave enough choices. to choose also −A4 , which is not compatible with B4,k . So we can also exclude C3 . In the remaining cases, for C1 and C2 , we do not have any other C’s at our disposal. Therefore we cannot complete to a sequence and so we can eliminate C1 and C2 . Altogether, we have removed now all C’s, and as we have seen above, it is not possible to complete to a strongly exceptional sequence of length 7.  Table 1. The cohomology free line bundles Name (c1 , c2 , c3 , c4 , c5 ) A1 (0, 0, 1, 0, 0) A2

(0, 1, 1, 0, 0)

A3

(0, 1, 2, 1, 0)

A4

(1, 1, 1, 0, 0)

A5

(1, 1, 2, 1, 0)

A6

(1, 2, 2, 1, 0)

A7

(1, 2, 3, 1, 0)

Compatible with −A6 , A7 , −C2 , C3 , −C7 , C8 , −C10 , C11 , −C6 −B1,k , B2,k , −B3,k , B4,k , −B6,k , B7,k −A5 , A7 , −C1 , C3 , −C4 , C5 , −C7 , −C8 , C10 , C11 −B1,k , B3,k , −B2,k , B4,k , −B5,k , B7,k −A4 , A7 , −C1 , −C3 , C4 , C5 , C6 −B2,k , B5,k , −B3,k , B6,k , −B4,k , B7,k if k ≥ 1 −A3 , A7 , −C1 , −C2 , −C3 , C8 , C10 , C11 , B2,k if k ≥ 2, B3,k if k ≥ 2, B4,k , −B5,k , −B6,k , −B7,k −A2 , A7 , −C1 , C9 , B1,k , B2,k if k ≥ 2, −B3,k , −B4,k , B5,k , −B6,k −A1 , A7 , B1,k , −B2,k , B3,k if k ≥ 2, −B4,k , B6,k , −B7,k A1 , A2 , A3 , A4 , A5 , A6 , B1,k for k ≥ 3, B7,1 , Bi,k for i = 2, . . . , 7 and k ≥ 2, −Bi,k for i = 1, . . . , 7.

22

CHAPTER 2. A COUNTEREXAMPLE TO KING’S CONJECTURE

Name B1,k , k ≥ 2 B2,k , k ≥ 1 B3,k , k ≥ 1 B4,k , k ≥ 1 B5,k , k ≥ 1 B6,k , k ≥ 1 B7,k , k ≥ 0

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11

(c1 , c2 , c3 , c4 , c5 ) (k, 2k − 1, 3k − 2, k, 1)

Compatible with −A1 , −A2 , A5 , A6 , A7 if k ≥ 3, −A7 , B1,k−1 if k ≥ 2, B1,k+1 , B2,k−1 , B2,k , B3,k−1 , B3,k (k, 2k − 1, 3k − 1, k, 1) A1 , −A2 , −A3 , A4 if k ≥ 2, A5 −A6 , A7 if k ≥ 2, −A7 , B1,k if k ≥ 2, B1,k+1 , B2,k−1 if k ≥ 2, B2,k+1 , B4,k−1 if k ≥ 2, B4,k , B5,k−1 if k ≥ 2, B5,k (k, 2k, 3k − 1, k, 1) −A1 , A2 , −A3 , A4 if k ≥ 2, −A5 , A6 if k ≥ 2, A7 if k ≥ 2, −A7 , B1,k if k ≥ 2, B1,k+1 , B3,k−1 if k ≥ 2, B3,k+1 B4,k−1 if k ≥ 2, B4,k , B6,k−1 if k ≥ 2, B6,k (k, 2k, 3k, k, 1) A1 , A2 , −A3 , A4 if k ≥ 2, −A5 , −A6 , A7 if k ≥ 2, −A7 , B2,k , B2,k+1 , B3,k , B3,k+1 , B4,k−1 if k ≥ 2, B4,k+1 , B7,k−1 if k ≥ 2, B7,k (k, 2k, 3k + 1, k + 1, 1) −A2 , A3 , −A4 , A5 if k ≥ 2, A7 if k ≥ 2, −A7 , B2,k , B2,k+1 , B5,k−1 if k ≥ 2, B5,k+1 , B7,k−1 if k ≥ 2, B7,k (k, 2k + 1, 3k + 1, k + 1, 1) −A1 , A3 , −A4 , −A5 , A6 if k ≥ 2, A7 if k ≥ 2, −A7 , B3,k , B3,k+1 , B6,k−1 if k ≥ 2, B6,k+1 , B7,k−1 if k ≥ 2, B7,k (k, 2k + 1, 3k + 2, k + 1, 1) A1 if k ≥ 1, A2 if k ≥ 1, A3 if k ≥ 1, −A4 , −A5 , −A6 , A7 if k ≥ 1, −A7 , B4,k , B4,k+1 , B5,k , B5,k+1 , B6,k , B6,k+1 , B7,k−1 if k ≥ 2, B7,k+1 (2, 4, 7, 3, 2) −A2 , −A3 , −A4 , −A5 , C3 , C4 , C8 , C9 , B2,1 , B2,2 , B4,1 , B5,1 , B7,0 , B7,1 (2, 5, 7, 3, 2) −A1 , A3 , −A4 , C3 , C10 , C6 B3,1 , B3,2 , B4,1 , B6,1 , B7,0 , B7,1 (2, 5, 8, 3, 2) A1 , A2 , −A3 , −A4 , C1 , C2 , C5 , C11 , B4,1 , B4,2 , B7,0 , B7,1 (2, 5, 9, 4, 2) −A2 , A3 , C1 , C5 , B5,1 , B5,2 , B7,0 , B7,1 (2, 6, 10, 4, 2) A1 , A2 , A3 , C3 , C4 , C6 , B7,0 , B7,2 (2, 6, 9, 4, 2) −A1 , A3 , C2 , C5 , B6,1 , B6,2 , B7,0 , B7,1 (3, 5, 7, 3, 2) −A1 , −A2 , C8 , C10 , B1,2 , B2,1 , B2,2 , B3,1 , B3,2 , B4,1 (3, 5, 8, 3, 2) A1 , −A2 , A4 , C1 , C7 , C11 B2,1 , B2,2 , B4,1 , B4,2 (3, 5, 9, 4, 2) A5 , C1 , B2,1 , B2,2 , B5,1 , B5,2 (3, 6, 8, 3, 2) −A1 , A2 , A4 , C2 , C7 , C11 B3,1 , B3,2 , B4,1 , B4,2 (3, 6, 9, 3, 2) A1 , A2 , A4 , C3 , C8 , C10 , B4,1 , B4,2

CHAPTER 3

Divisorial cohomology vanishing on toric varieties 3.1. Introduction The goal of this paper is to more thoroughly understand cohomology vanishing for divisorial sheaves on toric varieties. The motivation for this comes from the calculations of [HP06], where a counterexample to a conjecture of King [Kin97] concerning the derived category D b (X) of smooth complete toric varieties was presented. Based on work of Bondal (see [Rud90], [Bon90]), it was conjectured that on every smooth complete toric variety X there exists a full strongly exceptional collection of line bundles. That is, a collection of line bundles L1 , . . . , Ln on X which generates Db (X) and has the property that Extk (Li , Lj ) = 0 for all k > 0 and L all i, j. Such a collection induces an equivalence of categories RHom( i Li , . ) : Db (X) −→  L D b End( i Li ) − mod . This possible generalization of Beilinson’s theorem (pending the existence of a full strongly exceptional collection) has attracted much interest, notably also in the context of the homological mirror conjecture [Kon95]. For line bundles, the problem of Ext-vanishing can be reformulated to a problem of cohomology vanishing for line bundles by the isomorphisms Extk (Li , Lj ) ∼ = H k (X, Liˇ ⊗ Lj ) = 0 for all k ≥ 0 and all i, j. So we are facing a quite peculiar cohomology vanishing problem: let n denote the rank of the Grothendieck group of X, then we look for a certain constellation of n(n − 1) – not necessarily distinct – line bundles, all of which have vanishing higher cohomology groups. The strongest general vanishing theorems so far are of the Kawamata-Viehweg type (see [Mus02] and [Fuj07], and also [Mat02a] for Bott type formulas for cohomologies of line bundles), but it can be seen from very easy examples, such as Hirzebruch surfaces, that these alone in general do not suffice to prove or disprove the existence of strongly exceptional collections by means of cohomology vanishing. In [HP06], on a certain toric surface X, all line bundles L with the property that H i (X, L) = H i (X, Lˇ) = 0 for all i > 0 were completely classified by making use of a explicit toric representation of the cohomology vanishing problem for line bundles. This approach exhibits quite complicated combinatorial as well as number theoretic conditions for cohomology vanishing which we are going to describe in general. We will consider and partially answer the following more general problem. Let D be a Weil divisor on any toric variety X and V ⊂ X a torus invariant closed subscheme, then what are necessary and sufficient conditions for the (global) local cohomology modules HVi X, OX (D) to vanish? Given this spectrum of cohomology vanishing problems, we have on one extreme end the cohomology vanishing problem for line bundles, and on another extreme end the classification problem for maximal Cohen Macaulay (MCM) modules over semigroup rings: on an affine  toric variety X, the sheaf OX (D) is MCM if and only if the local cohomologies Hxi X, OX (D) vanish for i 6= dim X, where x ∈ X is the torus fixed point. These local cohomologies have been studied by Stanley [Sta82], [Sta96] (see also [Van92] for generalizations), and Bruns and Gubeladze [BG03] showed that only finitely many sheaves in this class are MCM. MCM sheaves over affine toric varieties have only been classified for some special cases (see, for instance [BGS87] 23

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CHAPTER 3. DIVISORIAL COHOMOLOGY VANISHING ON TORIC VARIETIES

and chapter 16 of [Yos90]). Our contribution will be to give a more explicit combinatorial characterization of MCM modules of rank one over normal semigroup rings and their ties to the birational geometry of toric varieties. One important aspect of our results is that, though we will also make use of Q-divisors, our vanishing results will completely be formulated in the integral setting. We will illustrate the effect of this by the following example.  Consider the weighted projective surface P(2, 3, 5). Then P(2, 3, 5) is isomorphic to Z and, after fixing the generator D = 1 the divisor class group A 1  of A1 P(2, 3, 5) to be Q-effective, the torus invariant irreducible divisors can be identified with the integers 2, 3, and 5, and the canonical divisor has class −10. By Kawamata-Viehweg we  obtain that H 2 (P(2, 3, 5), O(kD) = 0 for k > −10. However, as we will explain in more detail below, the set of all divisors kD with nontrivial second cohomology is given by all k with −k = 2r + 3s + 5t with r, s, t positive integers. So, Kawamata-Viehweg misses the divisor −11D. The reason is that the toric Kawamata-Viehweg vanishing theorem tells us that the cohomology of some divisor D′ vanishes if the rational equivalence class over Q of D′ − KP(2,3,5)  is contained in the interior of the nef cone in A1 P(2, 3, 5) Q . Over the integers, the domain of cohomology vanishing thus in general is larger than over Q. Below we will see that this is a general feature of cohomology vanishing, even for smooth toric varieties, as can be seen, for instance, by considering the strict transform of the divisor −11D along some toric blow-up X −→ P(2, 3, 5) such that X is smooth. The main results. The first main result will be an integral version of the Kawamata-Viehweg vanishing theorem. Consider the nef cone nef(X) ⊂ Ad−1 (X)Q , then the toric KawamataViehweg vanishing theorem (see Theorem 3.4.37) can be  interpreted such that if D − KX is contained in the interior of nef(X), then H i X, OX (D) = 0 for all i > 0. For our version we will define a set Anef ⊂ Ad−1 (X), which we call the arithmetic core of nef(X) (see definition 3.5.8). The set Anef has the property that it contains all integral Weil divisors which map to the interior of the cone KX + nef(X) in Ad−1 (X)Q . But in general it is strictly larger, as in the example above. We can lift the cohomology vanishing theorem for divisors in nef(X) to Anef :  Theorem (3.5.11): Let X be a complete toric variety and D ∈ Anef . Then H i X, OX (D) = 0 for all i > 0. One can consider Theorem 3.5.11 as an “augmentation” of the standard vanishing theorem for nef divisors to the subset Anef of Ad−1 (X). In general, Theorem 3.5.11 is slightly stronger than the toric Kawamata-Viehweg vanishing theorem and yields refined arithmetic conditions. However, the main goal of this paper is to find vanishing results which cannot directly be derived from known vanishing theorems. Let D be a nef Cartier divisor whose Iitaka dimension is positive but smaller than d. This class of divisors is contained in nonzero faces of the nef cone of X which are contained in the intersection of the nef cone with the boundary of the effective cone of X (see Section 3.5.1). Let F be such a face. Similarly as with Anef , we can define for the inverse cone −F an arithmetic core A−F (see 3.5.8) and associate to it a vanishing theorem, which may be considered as the principal result of this article:  Theorem (3.5.14): Let X be a complete d-dimensional toric variety. Then H i X, O(D) = 0 for every i and all D which are contained in some A−F , where F is a face of nef(X) which cointains nef divisors of Iitaka dimension 0 < κ(D) < d. If A−F is nonempty, then it contains infinitely many divisor classes.

3.1. INTRODUCTION

25

This theorem cannot be an augmentation of a vanishing theorem for −F , as it is not true in general that H i X, OX (−D) = 0 for all i for D nef of Iitaka dimension smaller than d. In particular, the set of Q-equivalence classes of elements in A−F does not intersect −F . For the case of a toric surface X we show that above vanishing theorems combine to a nearly complete vanishing theorem for X. Recall that in the fan associated to a complete toric surface X every pair of opposite rays by projection gives rise to a morphism from X to P1 (e.g. such a pair does always exist if X is smooth and X 6= P2 ). Correspondingly, we obtain a family of nef divisors of Iitaka dimension 1 on X given by the pullbacks of the sheaves OP1 (i) for i > 0. We get: Theorem (3.5.18): Let X be a complete toric surface. Then there are only finitely S many i divisors D with H X, OX (D) = 0 for all i > 0 which are not contained in Anef ∪ F A−F , where the union ranges over all faces of nef(X) which correspond to pairs of opposite rays in the fan associated to X. Some more precise numerical characterizations on the sets A−F will be given in subsection 3.5.1. The final result is a birational characterization of MCM-sheaves of rank one. This is a test case to see whether point of view of birational geometry might be useful for classifying more general MCM-sheaves. The idea for this comes from the investigation of MCM-sheaves over surface singularities in terms of resolutions in the context of the McKay correspondence (see [GSV83], [AV85], [EK85]). For an affine toric variety X, in general one cannot expect to find a similar nice correspondence. However, there is a set of preferred partial resolutions of ˜ −→ X which is parameterized by the secondary fan of X. Our result is a singularities π : X toric analog of a technical criterion of loc. cit. Theorem (3.5.31): Let X be a d-dimensional affine toric variety whose associated cone has simplicial facets and let D ∈ Ad−1 (X). If Ri π∗ OX˜ (π ∗ D) = 0 for every regular triangulation ˜ −→ X, then OX (D) is MCM. For d = 3 the converse is also true. π:X Note that the facets of a 3-dimensional cone are always simplicial. To prove our results we will require a lot of bookkeeping, combining various geometric, combinatorial and arithmetic aspects of toric varieties. This has the unfortunate effect that the exposition will be rather technical and incorporate many notions (though not much theory) coming from combinatorics. As this might be cumbersome to follow for the more geometrically inclined reader, we will give an overview of the key structures and explain how they fit together. From now X denotes an arbitrary d-dimensional toric variety, ∆ the fan associated to X, M∼ = Zd the character group of the torus which acts on X. We denote [n] := {l1 , . . . , ln } the set of primitive vectors of the 1-dimensional cones in ∆ and D1 , . . . , Dn the corresponding torus invariant prime divisors on X. Cohomology and simplicial complexes. We will follow the standard approach for computing cohomology of torus invariant Weil divisors, using the induced eigenspace decomposition. Let D be such a divisor on X and V ⊆ X a torus-invariant subscheme, then: M   HVi X, OX (D) ∼ HVi X, OX (D) m . = m∈M

ˆ the simplicial model of ∆, i.e. the abstract simplicial complex on the set [n] such We denote ∆ ˆ iff there exists a cone σ in ∆ such that elements in I are faces of that any subset I ⊂ [n] is in ∆ ˆ V , by considering only those cones in ∆ whose associated σ. Similarly, we define a subcomplex ∆ orbits in X are not contained in V (see also Section 3.2.1). For a given torus invariant divisor

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CHAPTER 3. DIVISORIAL COHOMOLOGY VANISHING ON TORIC VARIETIES

P ˆ m and ∆ ˆ V,m the full subcomplexes which D = ni=1 ci Di and any character m ∈ M , we set ∆ are supported on those li with li (m) < −ci . Now the general formula for cohomology of OX (D) ˆ and ∆ ˆ V with coefficients in our is given as the relative reduced cohomology of the complexes ∆ base field k: Theorem (3.2.11): Let D be a torus invariant Weil divisor on X. Then for every torus invariant subscheme V of X, every i ≥ 0 and every m ∈ M :  ˆ m, ∆ ˆ V,m ; k). HVi X, OX (D) ∼ = H i−1 (∆ m

This theorem is an easy consequence of the characterization for the case V =  standard ˆ m ; k). We state it explicitly for X ([EMS00]), which says that H i X, OX (D) m ∼ = H i−1 (∆ reference purposes, as it encompasses both, the case of global and local cohomology. D = Pn The circuit geometry of a i toric variety. By above theorem, for an invariant divisor ˆ ∆ ˆ V as c D , the eigenspaces H X, O (D) depend on the simplicial complexes ∆, i i X i=1 V m c well as on the position of the characters m with respect to the hyperplanes Hi = {m ∈ MQ | c li (m) = −ci }, where MQ = M ⊗Z Q. The chamber decomposition of MQ induced by the Hi (or their intersection poset) can be interpreted as the combinatorial type of D. Our strategy will be to consider the variations of combinatorial types depending on c = (c1 , . . . , cn ) ∈ Qn . The solution to this discriminantal problem is given by the discriminantal arrangement associated to the vectors l1 , . . . , ln , which has first been considered by Crapo [Cra84] and Manin and Schechtman [MS89]. The discriminantal arrangement is constructed as follows. Consider the standard short exact sequence associated to X: L

G

0 −→ MQ −→ Qn −→ AQ −→ 0,  where L is given by L(m) = l1 (m), . . . , ln (m) , and AQ := Ad−1 (X) ⊗Z Q is the rational divisor class group of X. The matrix G is called the Gale transform of L, and its i-th column Di is the Gale transform of li . The most important property of the Gale transform is that the linear dependencies among the li and among the Di are inverted. That is, for any subset among the li which forms a basis, the complementary subset of the Di forms a basis of AQ , and vice versa. Moreover, for every circuit, i.e. a minimal linearly dependent subset, C ⊂ [n] the complementary set {Di | li ∈ / C} spans a hyperplane HC in AQ . Then the discriminantal arrangement is given by the hyperplane arrangement

(2)

{HC | C ⊂ [n] circuit}. The stratification of AQ by this arrangement then is in bijection with the combinatorial types c of the arrangements given by the Hi under variation of c. As we will see, virtually all properties of X concerning its birational geometry and cohomology vanishing of divisorial sheaves on X depend on the discriminantal arrangement. In particular, (see Proposition 3.4.18), the discriminantal arrangement coincides with the hyperplane arrangement generated by the facets of the secondary fan. Ubiquitous standard constructions such as the effective cone, nef cone, and the Picard group can easily be identified as its substructures. Another interesting aspect is that the discriminantal arrangement by itself (or the associated matroid, respectively) represents a combinatorial invariant of the variety X, which one can refer to as its circuit geometry. This circuit geometry refines the combinatorial information coming with the toric variety, that is, the fan ∆ and the matroid structure underlying the li (i.e. their linear dependencies). It depends only on the li , and even for two combinatorially equivalent fans ∆, ∆′ such that corresponding sets of primitive vectors l1 , . . . , ln and l1′ , . . . , ln′ have the same underlying linear dependencies, their associated circuit geometries are different in general.

3.1. INTRODUCTION

27

This already is the case for surfaces, see, for instance, Crapo’s example of a plane tetrahedral line configuration ([Cra84], §4). Falk ([Fal94], Example 3.2) gives a 3-dimensional example. Toric 1-circuit varieties and the diophantine Frobenius problem. A special class of simplicial toric varieties, which we call toric 1-circuit varieties are those with primitive vectors l1 , . . . , ld+1 forming a unique circuit. Such a circuit comes with a relation (3)

d+1 X

αi li = 0

i=1

where the αi are nonzero integers whose largest common divisor is one. This relation is unique up to sign and we assume for simplicity that αi > 0 for at least one i. For a relation as in (3), we denote P(α1 , . . . , αd+1 ) the toric variety whose fan is generated by maximal cones spanned by the sets {lj | i 6= j} for every i with αi > 0. Given that at least one αi < 0, we can likewise consider the variety P(−α1 , . . . , −αd+1 ) and it is not difficult to see that these are the only two simplicial fans supported on the primitive vectors l1 , . . . , ld+1 . The integers α1 , . . . , αd+1 determine P(α1 , . . . , αd+1 ) uniquely up to a quotient by a finite group (which we suppress for this exposition). In particular, if αi > 0 for all i, then the toric circuit variety is isomorphic to the weighted projective space with weights αi . If αi < 0 for at least one i, the associated toric 1-circuit variety is a local model for a flip (or flop) P(α1 , . . . , αd+1 ) _ _ _// P(−α1 , . . . , −αd+1 ). This kind of operation shows up in the toric minimal model program and has been well-studied (see [Rei83]), and relations of type (3) play an important role for the classification of toric varieties (see [Oda88], §1.6, for instance). The reason for studying toric 1-circuit varieties in isolation is that they are the building blocks for our arithmetic conditions on cohomology vanishing. Let P(α1 , . . . , αd+1 ) denote a weighted projective space with reduced weights αi and D the unique Q-effective generator of  Ad−1 P(α1 , . . . , αd+1 ) . Then by a standard construction we get  dim H 0 P(α1 , . . . , αd+1 ), OP(α1 ,...,αd+1 ) (nD) d+1 X d+1 ki αi = n} =: VPα1 ,...,αd+1 (n), = {(k1 , . . . , kd+1 ) ∈ N | i=1

for every n ∈ Z, where VPα1 ,...,αd+1 is the so-called vector partition function or denumerant function with respect to the αi . The problem of determining the zero set of VPα1 ,...,αd+1 (or the maximum of this set) is quite famously known as the diophantine Frobenius problem. This problem is hard in general (though not necessarily in cases of practical interest) and there does not exist a general closed expression to determine the zero set. For a survey of the diophantine Frobenius problem we refer to the book [Ram05]. For more general toric 1-circuit varieties and higher cohomology groups, the cohomology vanishing problem can be expressed in similar terms. Accepting the fact that the general cohomology vanishing problem for toric varieties is at least as hard as the diophantine Frobenius problem it still makes sense to simplify the general situation by lifting vanishing conditions from the 1-circuit case in a suitable way. In subsection 3.3 we will cover the cohomology vanishing problem for toric 1-circuit varieties somewhat more extensively than it would strictly be necessary for proving our main theorems. The reason is that, from the general setup we have to provide, we can derive essentially for free a complete characterization of divisorial cohomology vanishing for these varieties. It should be instructive to see this kind of results explicitly for the simplest possible cases. As the class of toric 1-circuit varieties contains the weighted projective spaces, our treatment can be considered as toric supplement to standard references such as [Del75], [Dol82], [BR86] and should also serve as a reference.

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Lifting from the 1-circuit case to the general case. The basic idea here is to transport the discriminantal arrangement from AQ to some diophantine analog in Ad−1 (X). For any circuit C ⊂ [n] there is a short exact sequence 0 −→ HC −→ AQ −→ AC,Q −→ 0. By choosing one of the two possible simplicial fans supported on C (which needs not necessarily be realized as a subfan of ∆), we have an induced orientation on HC and we can identify AC,Q = AC ⊗Z Q ∼ = Q with the group of Q-divisors on the corresponding 1-circuit variety. By lifting the surjection AQ → AC,Q to its integral counterpart Ad−1 (X) → AC , we lift the zero set of the corresponding vector partition function on AC to Ad−1 (X). By doing this for every circuit C, we construct in Ad−1 (X) what we call the Frobenius discriminantal arrangement. One can consider the Frobenius discriminantal arrangement as an arithmetic thickening of the discriminantal arrangement. This thickening in general is just enough to enlarge the relevant strata in the discriminantal arrangement such that it encompasses the Kawamata-Viehweg-like theorems. To derive other vanishing results, our analysis will mostly be concerned with analyzing the birational geometry of X and its implications on the combinatorics of the discriminantal arrangement, and the transport of this analysis to the Frobenius arrangement. Overview. Section 3.2 is devoted to the proof of theorem 3.2.11. In section 3.3 we give a complete characterization of cohomology vanishing for toric 1-circuit varieties. In section 3.4 we survey discriminantal arrangements, secondary fans, and rational aspects of cohomology vanishing. Several technical facts will be collected which are important for the subsequent sections. Section 3.5 contains all the essential results of this work. In 3.5.1 we will prove our main arithmetic vanishing results. These will be applied in 3.5.2 to give a quite complete characterization of cohomology vanishing for toric surfaces. Section 3.5.3 is devoted to maximally Cohen-Macaulay modules. Acknowledgments. Thanks to Laurent Bonavero, Michel Brion, Lutz Hille, Vic Reiner, and Jan Stienstra for discussion and useful hints. 3.2. Cohomology of Divisorial Sheaves 3.2.1. Toric Preliminaries. In this section we first introduce notions from toric geometry which will be used throughout the rest of the paper. As general reference for toric varieties we use [Oda88], [Ful93]. We will always work over an algebraically closed ground field k. Fans and combinatorics. Let ∆ be a fan in the rational vector space NQ := N ⊗Z Q over a lattice N ∼ = Zd . Let M be the lattice dual to N , then the elements of N represent linear forms on M and we write n(m) for the canonical pairing M × N → Z, where n ∈ N and m ∈ M . This pairing extends naturally over Q, MQ × NQ → Q. Elements of M are denoted by m, m′ , etc. if written additively, and by χ(m), χ(m′ ), etc. if written multiplicatively, i.e. χ(m + m′ ) = χ(m)χ(m′ ). The lattice M is identified with the group of characters of the algebraic torus T = Hom(M, k∗ ) ∼ = (k∗ )d which acts on the toric variety X = X∆ associated to ∆. Moreover, we will use the following notation: • cones in ∆ are denoted by small greek letters ρ, σ, τ, . . . , their natural partial order by ≺, i.e. S ρ ≺ τ iff ρ ⊆ τ ; • |∆| := σ∈∆ σ denotes the support of ∆; • for 0 ≤ i ≤ d we denote ∆(i) ⊂ ∆ the set of i-dimensional cones; for σ ∈ ∆, we denote σ(i) the set of i-dimensional faces of σ; • Uσ denotes the associated affine toric variety for any σ ∈ ∆; • σ ˇ := {m ∈ MR | n(m) ≥ 0 for all n ∈ σ} is the cone dual to σ; • σ ⊥ = {m ∈ MR | n(m) = 0 for all n ∈ σ};

3.2. COHOMOLOGY OF DIVISORIAL SHEAVES

29

• σM := σ ˇ ∩ M is the submonoid of M associated to σ; ⊥ := σ ⊥ ∩ M is the unique maximal subgroup of M contained in σ ; • σM M L • the monoid ring k[σM ] ∼ = m∈σM k · χ(m) is identified with the coordinate ring of Uσ , and in particular, the group ring k[M ] with the coordinate ring of T = U0 . We will mostly be interested in the structure of ∆ as a combinatorial cellular complex. For this, we make a few convenient identifications. We always denote n the cardinality of ∆(1). i.e. the number of 1-dimensional cones (rays) and [n] := {1, . . . , n}. The primitive vectors along rays are denoted l1 , . . . , ln , and usually we will identify the sets ∆(1), the set of primitive vectors, and [n]. Also, we will often identify σ ∈ ∆ with the set σ(1) ⊂ [n]. With these identifications, and using the order of [n], we obtain a combinatorial cellular complex with support [n]; we may consider this complex as a combinatorial model for ∆. In the case where ∆ is simplicial, this complex is just a combinatorial simplicial complex in the usual sense. If ∆ is not simplicial, ˆ of ∆, modelled on [n]: some subset I ⊂ [n] is in ∆ ˆ iff there we consider the simplicial cover ∆ exists some σ ∈ ∆ such that I ⊂ σ(1). The identity on [n] then induces a surjective morphism ˆ −→ ∆ of combinatorial cellular complexes. This morphism has a natural representation in ∆ ˆ with the fan in Qn which is defined as follows. Let e1 , . . . , en terms of fans. We can identify ∆ n the standard basis of Q , then for any set I ⊂ [n], the vectors {ei }i∈I span a cone over Q≥0 iff ˆ is open in An , and the vector there exists σ ∈ ∆ with I ⊂ σ(1). The associated toric variety X k ˆ → ∆. space homomorphism defined by mapping ei 7→ li for i ∈ [n] induces a map of fans ∆ ˆ → X is the quotient presentation due to Cox [Cox95]. We will not The induced morphism X make explicit use of this construction, but it may be useful to have it in mind. Weil divisors and divisorial sheaves. An important fact used throughout this work is the following exact sequence: (4)

L

M −→ Zn −→ Ad−1 (X) −→ 0

where L(m) = (l1 (m), . . . , ln (m)), i.e. as a matrix, the primitive vectors li represent the row vectors of L. Note that L is injective iff ∆ is not contained in a proper subspace of NQ . The sequence implies that P every Weil divisor D on X is rationally equivalent to a T -invariant Weil divisor, i.e. D ∼ ni=1 ci Di , where c = (c1 , . . . , cn ) ∈ Zn and D1 , . . . , Dn , the T -invariant irreducible divisors of X. Moreover, any two T -invariantPdivisors D, D′ are rationally equivalent if and only if there exists m ∈ M such that D − D′ = ni=1 li (m)Di . To every Weil divisor D, one associates its divisorial sheaf OX (D) = O(D) (we will omit the subscript X whenever there is no ambiguity), which is a reflexive sheaf of rank one, locally free if and only if D is Cartier. Recall that rational equivalence classes of Weil divisors are in bijection with isomorphism classes of divisorial sheaves. If D is T -invariant, the sheaf O(D) acquires a T -equivariant structure and the equivariant isomorphism classes of sheaves O(D) are one-to-one with Zn . For some σ ∈ ∆, the T -equivariant structure of O(D)|Uσ is equivalent to an M -graded structure over k[σM ] of the module M   Γ Uσ , O(D) m . Γ Uσ , O(D) ∼ = m∈M

To describe this M -graded structure more precisely, we introduce some more notation: P Definition 3.2.1: Let D = ni=1 ci Di be a T -invariant Weil divisor, then we set:  • RσD := Γ Uσ , O(D) for all σ ∈ ∆; c • HiD := Hi := {m ∈ MQ | li (m) ≥ −ci } and MiD := HiD ∩ M for all i ∈ [n]; T • MσD := i∈σ(1) MiD for all σ ∈ ∆; L • k[MσD ] := m∈MσD k · χ(m) as graded k[σM ]-submodule of k[M ].

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There is the following characterization which is quite standard, see, e.g., [Per04a]: Lemma 3.2.2: There is an isomorphism of graded k[σM ]-modules: RσD ∼ = k[MσD ]. Orbits and monomial sheaves. Some more notation: • let σ ∈ ∆, then orb(σ) denotes the orbit associated to σ in X, and V (σ) = orb(σ) the orbit closure; • let σ ∈ ∆, then Nσ is N intersected with the subvector space of NQ spanned by σ over Q, Mσ is the dual Z-module of Nσ , where there is a canonical identification ⊥ ; moreover, there is a natural isomorphism M ∼ σ ⊥ × M ; Mσ = M/σM = M σ • for any set F of faces of σ, we call the set star(F ) := {η | τ ≺ η for some τ ∈ F } the star of F ; if star(F ) = F , then F is called star-closed. Recall that for any σ ∈ ∆, dualizing σ via τ 7→ τ ⊥ ∩ σ ˇ induces an order-reversing one-to-one correspondence between faces of σ and faces of σ ˇ. Consider V ⊂ X a T -invariant closed subscheme and denote IV ⊂ OX its ideal sheaf. Locally, for every σ ∈ ∆, the sheaf IV corresponds to a monomial ideal BV,σ := Γ(Uσ , IV ), i.e. BV,σ ⊂ k[σM ] is generated by a finite set of characters χ(m) ∈ k[σM ]. For every pair τ ≺ σ, there exists an element mτ ∈ σ ⊥ ∩ τM such that τM = σM + Z≥0 (−mτ ), and Bτ = (Bσ )χ(mτ ) , the localization of Bσ at χ(mτ ). Conversely, every system {Bσ }σ∈∆ , which is compatible in this way, defines the ideal sheaf of a T -invariant closed subscheme of X. The reduced scheme Vred is a finite union of orbits, i.e. a Vred = orb(σ), where supp(V ) := {σ ∈ ∆ | BV,σ 6= k[σM ]}. σ∈supp(V )

We call supp(V ) the support of V in ∆. Note that supp(V ) is star-closed. Definition 3.2.3: Let I ⊂ [n], then we denote σI the minimal cone σ ∈ ∆ such that I ⊂ σ(1). Let F be any set of cones of ∆, then we set ˆ F := {I ⊂ [n] | σI ∈ ∆ / star(F )}, ˆ If F = {σ} for some σ ∈ ∆, then we write ∆ ˆ σ instead of which is a simplicial subcomplex of ∆. ˆ ˆ V instead ∆{σ} . If V ⊂ X is a T -invariant subscheme and F = supp(V ), then we also write ∆ ˆ supp(V ) , which we call the cosupport of V in ∆. ˆ of ∆ 3.2.2. ΓV -Acyclic Sheaves. Given a T -invariant Weil divisor D and a T -invariant closed subscheme V of X, we construct a class of sheaves on X which are acyclic with respect to the functor ΓV of sections with support in Vred . The affine case. Let X = Uσ be an affine toric variety with cone σ ⊂ NQ , thus [n] = σ(1). Definition 3.2.4: Let I ⊂ [n] be a nonempty subset, then we set [ ZID := M \ MiD i∈[n]

and define

k[ZID ] := coker

M i∈I

 ξ k[MiD ] −→ k[M ] ,

P where ξ = i ξi and ξi : k[MiD ] ֒→ k[M ] the canonical inclusions. In the special case where I = {i}, we write ZiD and k[ZiD ], respectively.

3.2. COHOMOLOGY OF DIVISORIAL SHEAVES

T

31

L ZiD and k[ZID ] ∼ = m∈ZID k · χ(m) as k[σM ]-modules. The D ZI are intersections of M with unbounded polyhedra whose recession cones point “backward” compared to the cone σ ˇ , as they are contained in the complementary half spaces of the HiD . In the special case where D is rationally equivalent to a character m ∈ M and I = [n], the k[ZID ] coincide with the usual injective modules in the category of M -graded k[σM ]-modules (see [GW78] and [MS04], §11). For I = [n], the modules k[ZID ] are Matlis duals of reflexive P ′ modules RσD , where D′ = −D + ni=1 Di (for graded Matlis duality, see also [MS04], §11). ′ D Let ρ ∈ [n] and m ∈ ZρD , then for any subset M ′ ⊂ ρ⊥ M the set m + M is contained in Zρ . This is in particular true when M ′ is a subgroup of ρ⊥ M . Thus the following lemma is immediate: Clearly, we have ZID =

i∈I

Lemma 3.2.5: Let M ′ ⊂ M be any subgroup and I ⊂ [n]. Then for any m ∈ ZID the set T m + M ′ is contained in ZID if and only if M ′ ⊂ ρ∈I ρ⊥ M.

We have the following properties for k[ZID ] with respect to local cohomology with torus invariant support: Proposition 3.2.6: Let V ⊂ Uσ be a T -invariant closed subscheme, D a T -invariant Weil divisor, and I ⊆ [n], then: ( ˆV k[ZID ] if I ∈ /∆ (i) ΓV k[ZID ] = 0 else. D (ii) The module k[ZI ] is ΓV -acyclic. The proof follows the lines of [TH86]: Proof. Denote B the monomial ideal such that V = spec(k[σM ]/B). Let first I ⊂ [n], such that τI ∈ / supp(V ). So there exists an integral element m in the relative interior of τI⊥ ∩ σ T ⊥ such that the monomial χ(m) is contained in B. As the group M ′ := ρ∈I ρ⊥ M contains τI ∩ σM (note that τI⊥ ∩ σM = M ′ ∩ σM ) and thus m ∈ M ′ , we have m + M ′ = M ′ . This implies that any power of χ(m) is a nonzero divisor of the module k[ZID ], and moreover, multiplication by χ(m) even represents an automorphism of the module k[ZID ]. So, any power of χ(m) also acts as an automorphism on the local cohomology modules HVi k[ZID ] for every i ≥ 0. But because the (ring theoretic) support of k[ZID ] is contained in V , for every element x ∈ HVi k[ZID ] there ˆV, exists some n > 0 such that χ(m)n x = 0, hence HVi k[ZID ] = 0 for every i ≥ 0. So for I ∈ ∆ (i) and (ii) are true. ˆ V , i.e τI ∈ supp(V ). In that case, B contains no monomial Now consider the case I 6⊂ ∆ ⊥ whose degree is contained in τI ∩ σM , and thus not in M ′ , so for every x ∈ k[ZID ], there exists some n > 0 such that B n x = 0. So, the support of k[ZID ] is contained in the support of B, and thus ΓV k[ZID ] = k[ZID ] and HVi k[ZID ] = 0 for every i > 0.  ˆ The general case. Let X be any toric variety and D a T -invariant Weil divisor. Let I ∈ ∆ a simplex. Let σ := σI and denote iσ : Uσ ֒→ X the inclusion morphism. Definition 3.2.7: We denote ZID := iσ∗ k[ZID ], where by abuse of notation we have identified the divisor D with its restriction to Uσ and the k[σM ]-module k[ZID ] with its associated sheaf over Uσ .

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Clearly, ZID is a quasi-coherent sheaf on X. Proposition 3.2.8: Let ZID be defined as before. Then: (i) The sheaf theoretical support of ZID is equal to V (σI ). (ii) For any τ ∈ star(σI ) and any τ ≺ η, the restriction map Γ(Uη , ZID ) −→ Γ(Uτ , ZID ) is constant. (iii) Γ(X, ZID ) ∼ = k[ZID ] as Γ(X, OX )-modules. Proof. For any τ ∈ star(σI ), we have the identity Γ(Uτ , ZID ) = k[ZID ], and (i) and (ii) follow immediately. So, we have k[ZID ] = Γ(V (σI ), ZID ) = Γ(X, ZID ), and (iii) follows.  Note that ZID does not depend on the particular choice of σ, as long as σI ⊂ σ. Corollary 3.2.9: The sheaves ZID are acyclic with respect to ΓV and HV0 . Proof. The acyclicity with respect to HV0 follows directly from the local acyclicity shown in 3.2.6. Consequently, the spectral sequence  E2p,q = H p X, HVq (X, ZID ) ⇒ HVp+q (X, ZID )

(see [Gro68], Exp. 1, Th´eor`eme 2.6) degenerates and we have the equality  HVp (X, ZID ) = H p X, HV0 (X, ZID ) . Note that we have

HV0 (X, ZID ) =

(

ZID 0

if orb(σI ) ⊂ V else.

ˇ So it remains to show that the higher cohomologies of ZID vanish. For this, consider the Cech D complex of ZI with respect to the covering Uσ1 , . . . , Uσk , where σi runs over the maximal cones of star(σI ): k M M D ZID −→ ZID −→ · · · −→ ZID −→ 0, 0 −→ ZI −→ i=1

i 0. + Proof. Let m ∈ M and denote ∆+ m ⊂ [n] the set of rays such that li (m) ≥ 0 for all i ∈ ∆m , − + + and denote S ∆m := [n] \ ∆m . Assume that i ∈ ∆m and assume that li is contained in the convex ˇ As hull C of i∈∆− Q≥0 li . The set C is a cone in NQ , and m is contained in the interior of −C. m 0 0 ρ is in the interior of C, this implies that −Cˇ is outside the half space H , and H has empty i

i

3.3. TORIC 1-CIRCUIT VARIETIES

35

ˇ So li cannot be contained in C. So, for every m ∈ M , intersection with the interior of −C. d−1 ∼ ˆ ∆m = C ∩ S is contractible and the statement follows.  Corollary 3.2.14 gives an easy idea how to construct examples of toric varieties with |∆| nonconvex such that H i (X, OX ) 6= 0 for some i > 0. Example 3.2.15: Consider the four primitive vectors l1 = (1, 0, 0), l2 = (0, 1, 0), l3 = (−1, 1, 1), l4 = (0, 0, 1), spanning a three dimensional cone σ. Let the fan ∆ be generated by the faces of this cone except the maximal cone. Let m ∈ M such that li (m) < 0 for i = 1, . . . , 4, then the ˆ m ) = 1, and thus H 2 (X, OX ) 6= 0. associated reduced cohomology vanishes except for dim H 1 (∆ 2 As a k[σM ]-module, H (X, OX ) is Artinian, but not finitely generated and it is easy to see that it is isomorphic to the Matlis dual Homk (ωUσ , k) of the canonical module of k[σM ] (see [MS04], §11). 3.3. Toric 1-Circuit Varieties We now study divisorial cohomology vanishing for the simplest possible toric varieties which are not affine. Consider primitive vectors l1 , . . . , ln , which form a so-called circuit, i.e. a minimally linearly dependent set in N . Then there exists a relation X αi li = 0, i∈[n]

which is unique up to a common multiple of the αi , and the αi are nonzero. For simplifying the discussion, we will assume that the li generate a submodule N[n] of finite index in N , in particular, we have n = d + 1. Without loss of generality, we will assume that the α` i are integral + and gcd{|αi |}i∈[n] = 1. For a fixed choice of the αi , we have a partition [n] = C C− , where ± C = {i ∈ [n] | ±αi > 0}. This decomposition depends only on the signs of the αi ; flipping the signs exchanges C+ and C− . We want to keep track of this two possibilities and call ` the choice ` − + of C C =: C the oriented circuit with underlying circuit [n], and −C := −C+ −C− its inverse, where −C± := C∓ . The primitive vectors li can support at most two simplicial fans, each corresponding to an oriented circuit. For fixed orientation C, we denote ∆ = ∆C the fan whose maximal cones are generated by [n] \ {i}, where i runs over the elements of C+ . The only exception for this procedure is the case where C+ is empty, which we leave undefined. The associated toric variety X∆C is simplicial and quasi-projective. Definition 3.3.1: We call a toric variety X = X∆C associated to an oriented circuit a toric 1-circuit variety. Now let us assume that the sublattice N[n] which is generated by the li is saturated, i.e. N[n] ⊗Z Q ∩ N = N[n] . Then we have an exact sequence (5)

L

G

M −→ Zn −→ A −→ 0,

where L = (l1 , . . . , ln ) considered as a tuple of linear forms on M , A ∼ = Z and G = (α1 , . . . , αn ) a (1 × n)-matrix, i.e. we can consider the αi as the Gale transform of the li . Conversely, if the αi are given, then the li are determined up to a Z-linear automorphism of M . We will make more extensively use of the Gale transform later on. For generalities we refer to [OP91] and [GKZ94]. In the case that N[n] is not saturated, we can formally consider the inclusion of N[n] as the image of a saturated sublattice of an injective endomorphism ξ of N . The inverse images of the li with respect to ξ satisfy the same relation as the li . Therefore, a general toric circuit

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CHAPTER 3. DIVISORIAL COHOMOLOGY VANISHING ON TORIC VARIETIES

variety is completely specified by ξ and the integers αi . More precisely, a toric 1-circuit variety is specified by the Gale duals li of the αi and a an injective endomorphism ξ of N with the property that ξ(li ) is primitive in N for every i ∈ [n]. Definition 3.3.2: Let α = (α1 , . . . , αn ) ∈ Zn with αi 6= 0 for every i and gcd{|αi |}i∈[n] = 1, C the unique oriented circuit with C+ = {i | αi > 0}, and ξ : N −→ N an injective endomorphism of N which maps the Gale duals of the αi to primitive elements li in N . Then we denote P(α, ξ) the toric 1-circuit variety associated to the fan ∆C spanned by the primitive vectors ξ(li ). The endomorphism ξ translates into an isomorphism P(α, ξ) ∼ = P(α, idN )/H, where H ∼ = spec k[N/N[n] ]. Note that in positive characteristic, H in general is a group scheme rather than a proper algebraic group.  In sequence (5), we can identify A with the divisor class group Ad−1 P(α, idN ) . Sim∼ ilarly, we get A  d−1 P(α, ξ) = A ⊕ H and the natural surjection from Ad−1 P(α, ξ) onto Ad−1 P(α, idN ) just projects away the torsion part. The P(α, ξ) are an important building block for general toric varieties and therefore they will play a distinguished role in later sections. In fact, to every extremal curve V (τ ) in some simplicial toric variety X, there is associated some variety P(α, ξ) whose fan ∆C is a subfan of ∆ and P(α, ξ) and which embeds as an open invariant subvariety of X. If |C+ | ∈ / {n, n − 1}, the primitive vectors li span a convex polyhedral cone, giving rise to an affine toric variety Y and a canonical morphism π : P(α, ξ) −→ Y which is a partial resolution of singularities. Sign change α → −α then encodes the transition from C to −C and provides a local model for well-known combinatorial operation which called bistellar operation [Rei99] or modification of a triangulation [GKZ94]. In toric geometry usually it is also called a flip: flip P(α, ξ) _ _ _ _ _ _ _// P(−α, ξ)

FF FF π FF FF F##

v

π ′ vvv

Y

v vv vzz v

For |C+ | = d − 1, we identify P(−α, ξ) with Y and just obtain a blow-down. In the case αi > 0 for all i and ξ = idN , we just recover the usual weighted projective spaces. In many respects, the spaces P(α, ξ) can be treated the same way as has been done in the standard references for weighted projective spaces, see [Del75], [Dol82], [BR86]. In our setting there is the slight simplification that we naturally can assume that gcd{|αj |}j6=i = 1 for every i ∈ [n], which eliminates the need to discuss reduced weights.  Remark 3.3.3: On P(α, ξ) the sheaves O(D) for D ∈ Ad−1 P(α, ξ) can be constructed via the homogeneous coordinate ring. For sake of information we present the relevant data without proof and refer to [Cox95] for details (see also [Per04a]). The homogeneous coordinate ring L is given by the polynomial ring S := k[x , . . . , x ] with grading S = S 1 d+1 α∈A α , where A :=  Ad−1 P(α, ξ) and degA xi = [Di ] ∈ A. The grading is induced by the action of the group scheme + A˜ = spec k[A] on spec S = Ad+1 k . The irrelevant ideal B ⊂ S is of the form B = hxi | i ∈ C i ˜ By sheafification, every and the variety P(α, ξ) then is a good quotient of Ad+1 \ V (B) by A. k ] for α ∈ A. If A is torsion free, then A˜ ∼ divisorial sheaf is of the form S(α), = k∗ and the Z-grading is given by degZ xi = αi .

3.3. TORIC 1-CIRCUIT VARIETIES

37

3.3.1. Singularities and Picard group. In general, P(α, ξ) is not smooth and its singularities depend on the degree of the sublattices of N spanned by subsets of the li with respect to their saturations. Definition 3.3.4: Let l1 , . . . , ln ∈ N be primitive vectors and let I ⊂ [n]. Then we denote NI ¯I its saturation in N , and rI the index of the submodule of N spanned by the li with i ∈ I, N ¯I . NI in N Up to the global torsion relative to ξ, the structure of its singularities is encoded in α: ¯I /NI is cyclic and for every Lemma 3.3.5: Assume that ξ is an automorphism of N , then N proper subset I ⊂ [n], we have rI = gcd{|αi |}i∈[n]\I . Moreover, r{i} = 1 for every i ∈ [n]. P P Proof. As gcd{|αi |}i∈C = 1, the relation i∈C αi li = 0 is unique up to sign and i∈C\I αi li ¯I with lI = λ · l′ . Clearly, if λ 6= 1, =: lI ∈ NI . Let λ := gcd{|αi |}i∈C\I and denote lI′ ∈ N I ¯I is spanned by NI and then lI′ is not contained in NI . As L generates N , the submodule N ¯I /NI is cyclic and generated by the image ¯lI of l′ in N ¯I /NI and λ must be a lI′ . Thus N I multiple of the order of l¯I . But λ isPthe least multiple such that λlI′ ∈ NI , as otherwise there would exist an integral relation i∈C βi li = 0 with gcd{|βi |}i∈C = 1 different from the original one, contradicting its uniqueness. The last assertion follows from the fact that the li are primitive.  Recall that the maximal cones of ∆C are spanned by the complementary rays of i for every i ∈ C+ . Therefore: Corollary 3.3.6: The variety P(α, idN ) has cyclic quotient singularities of degree αi for every i ∈ C+ . In presence of a nontrivial ξ, there are some more factors to take into account: ¯I /NI |. If I = σ(1) for some σ ∈ ∆C, Definition 3.3.7: Let I ⊂ C, then we denote sI := rI−1 |N we write sI =: sσ , and for I = C we simply write s[n] =: s.   Note that s = | det ξ|. The group Pic P(α, ξ) embeds into Ad−1 ( P(α, ξ) as a subgroup of ∼ finite index. Therefore, if ξ is an automorphism, it follows that Pic  P(α, ξ) = Z. In general, ∼ by the isomorphism P(α, ξ) = P(α, idN )/H, the group Pic P(α, ξ) embeds into Pic P(α, idn ) as a subgroup of index s via pull-back. For simplicity, it is suitable to identify Pic P(α, ξ)  with its image in Pic P(α, idN ) . We have:  Proposition 3.3.8: Pic P(α, ξ) is generated by s · lcm{αi }i∈C+ .

 Proof. It suffices to prove that Pic P(α,  idN ) is generated by lcm{αi }i∈C+ . Let D = i∈C ci Di be the generator of Pic P(α, idN ) . Then D is specified by a collection {mi ∈ M | i ∈ C+ } such that li (mj ) = ci for all j and all j 6= i ∈ C. As changing the mi to mi + m for some m ∈ M just changes the linearization of O(D), but not the linear equivalence class of D, we can always assume that one of the mi is zero. This implies that cj =P 0 for all j 6= i, and li (mj ) = ci for all j 6= i and lk (mj ) = 0 for all k 6= i, j. Using the relation k∈C αk lk = 0, we thus obtain α ci = −li (mj ) = αji lj (mj ) for every j ∈ C+ . Therefore ci Di is a multiple of α−1 i · lcm{αj }j∈C+ · Di  and thus, by identifying αi with Di in Pic P(α, idN ) , a multiple of lcm{αj }j∈C+ . On the other  hand, it is easy to see that every such multiple yields a Cartier divisor on P(α, idN ). P

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 Remark 3.3.9: It follows  that a Cartierdivisor has no torsion part in Ad−1 P(α, ξ) , i.e. the embedding Pic  P(α, ξ) ֒→ Ad−1 P(α, ξ) factorizes through the section Ad−1 P(α, ξ) /H → Ad−1 P(α, ξ) ∼ = Z ⊕ H. As a consequence we have:

Corollary 3.3.10: Let i ∈ C+ , then the invariant prime divisor Di on P(α, ξ) is Cartier if and only if s = 1 and αj = 1 for every j ∈ C+ \ {i}. In particular, P(α, ξ) is smooth if and only if Di is Cartier for every i ∈ C+ . Proof. Let first s = 1 and αj = 1 for every j ∈ C+ \ {i}. Then by Corollary 3.3.6, the union of the Uσj , where j ∈ C+ \ {i}, is smooth and Di restricted to this union is Cartier. As Di is trivial on Uσi , it extends as Cartier divisor with trivial restriction on Uσi . Now let Di Cartier, then from proposition 3.3.8 we conclude Di = ( αsi lcm{αj }j∈C+ ) · Di , and thus s = 1 and αj = 1 for all i 6= j ∈ C+ .  3.3.2. General cohomology vanishing. In light of Theorem 3.2.11, for cohomology vanishing on a toric 1-circuit variety, we have to consider the reduced cohomology of simplicial complexes associated to its fan: ˆ C)I ; k) = 0 for all i. Moreover, Lemma 3.3.11: Let I ⊂ [n], such that I 6= C+ , then H i ((∆ + ˆ C)C+ ∼ (∆ = S |C |−2

and

− ˆ C)C− ∼ (∆ = B |C |−1 ,

where B k is the k-ball, with B −1 := ∅. ˆ C)C+ corresponds to the boundary of the (|C+ |− 1)-simplex, Proof. It is easy to see that (∆ + so it is homeomorphic to S |C |−2 . Similarly, {li }i∈C+ span a simplicial cone in ∆C and thus − ˆ C)C− ∼ (∆ = B |C |−1 . Now assume there exists i ∈ C+ \ I, then I is a face of the cone σi and ˆ C)I is contractible. On the other hand, if C+ is a proper subset of I, the set I ∩ C− spans a (∆ ˆ I then is homeomorphic to a simplicial decomposition cone τ in ∆C. The simplicial complex ∆ + ˆ C)C+ .  of the (|C | − 1)-ball with center τ and boundary (∆ By sequence (4),P the cohomology of a divisor D depends only on the choice of a T -invariant representative D = i∈[n] ci Di with ci ∈ Z. This choice is unique up to a twist by a character P in M , i.e. any divisor of the form i∈[n] li (m)Di for some m ∈ M is trivial. This can be interpreted more geometrically in terms of hyperplane arrangements in MQ . For a given choice of c = (c1 , . . . , cn ) ∈ Zn we set c

Hi := {m ∈ MQ | li (m) = −ci }. Then, replacing ci by ci + li (m) for some m ∈ M then corresponds to an integral translation of c the hyperplane arrangement {Hi }i∈[n] by −m. This hyperplane arrangement induces a chamber decomposition of MQ . If D ∼ 0, then the maximal chambers are all unbounded. If D ≁ 0, then we get precisely one additional chamber which is bounded (recall that in this section n = d + 1 and the li form a circuit). For this chamber, there are two possibilities. Either it is given by points m ∈ MQ such that li (m) ≤ −ci for i ∈ C− and li (m) ≥ −ci for i ∈ C− , or vice versa. Let us say in the first case that this chamber has signature C+ and in the second case it has signature C− (we will define signatures more generally in section 3.4.1). To determine cohomology vanishing we have to determine its signature and whether it contains lattice points. As already explained in the introduction, the number of lattice points in a bounded chamber is given by the vector partition function. Similarly, the number of lattice

39

3.3. TORIC 1-CIRCUIT VARIETIES

points m such that li (m) ≥ −ci for i ∈ C+ and li (m) < −ci for i ∈ C− coincides with the cardinality of the following set: X X {k1 , . . . , kd+1 ∈ Nd+1 | ki > 0 for i ∈ C− and ki Di − ki Di = D}. i∈C+

i∈C−

 The set of rational divisor classes in Ad−1 P(α, ξ) Q ∼ = Q corresponding to torus invariant divisors whose associated bounded chamber has signature either C+ or C− corresponds precisely  to the two open intervals (−∞, 0) and (0, ∞), respectively, in Ad−1 P(α, ξ) Q . In the integral case, we can consider arithmetic thickenings of these intervals as follows:  ξ) the complement of the semigroup of the P(α, Definition 3.3.12: We denote F ⊂ A C d−1 P P form i∈C− ci Di − i∈C+ ci Di , where ci ∈ N for all i with ci > 0 for i ∈ C+ .

The set FC is the complement of the set of classes whose associated chamber has signature C− and contains a lattice point. With this we can give a complete characterization of global cohomology vanishing:  Proposition 3.3.13: Let P(α, ξ) be as before with associated fan ∆C and D ∈ Ad−1 P(α, ξ) , then:  (i) H i P(α, ξ), O(D) = 0 for i 6= 0, |C+ | − 1;  + (ii) H |C |−1 P(α, ξ), O(D) = 0 iff D ∈ FC; (iii) if C+ 6= C, then H 0 P(α, ξ), O(D) 6= 0; (iv) if C+ = C, then H 0 P(α, ξ), O(D) = 0 iff D ∈ F−C. ˆ C)m , for m an Proof. The proof is immediate. Just observe that the simplicial complex (∆ ˆ C)C+ or (∆ ˆ C)C− . element in the bounded chamber, coincides either with (∆ 

Another case of interest is where C+ 6= C and V = V (τ ), where τ is the cone spanned by the li with i ∈ C− , i.e. V is the unique maximal complete torus invariant subvariety of P(α, ξ).  Proposition 3.3.14: Consider P(α, ξ) such that αi < 0 for at least one i, D ∈ Ad−1 P(α, ξ) and V the maximal complete torus invariant subvariety of P(α, ξ), then:  (i) HVd P(α, ξ), O(D) 6= 0;  |C− | (ii) HV P(α, ξ), O(D) = 0 iff D ∈ FC;  (iii) HVi P(α, ξ), O(D) = 0 for all i 6= d, |C− |.

ˆ C)I = ∆ ˆC ∼ ˆ C)V,I = (∆ ˆ C)V ∼ Proof. Consider first I = C, then (∆ (∆ = B d−1 and = S d−2 .    ˆ C, (∆ ˆ C)V ; k ∼ ˆ C)V ; k . As by ˆ C; k = 0 for all i and H i−1 ∆ It follows that H i ∆ = H i−2 (∆ assumption, C+ 6= C, so the associated hyperplane arrangement contains an unbounded chamber such that li (m) ≥ −ci for all i ∈ C and all m in this chamber. Hence (i) follows. As in the proof ˆ I is contractible whenever C+ ∩ I 6= ∅ and C− ∩ I 6= ∅. So in of lemma 3.3.11, it follows that ∆ ˆ I ) = 0 for all i and H i−1 (∆ ˆ I, ∆ ˆ V,I ; k) = H i−2 (∆ ˆ V,I ; k) for all i. that case H i (∆  + −2 + C i ˆ C)I = (∆C) ˆ V,I ∼ ˆ C)I , (∆C) ˆ V,I ; k = 0 for all i. For Now let I = C ; then (∆ , so H (∆ =S − − ˆ C)I ∼ ˆ C)V,I ∼ I = C− , then (∆ = B |C |−1 and (∆ = S |C |−2 , the former by Lemma 3.3.11, the latter ˆ C)V,I has empty intersection with star(τ ). This implies by Lemma 3.3.11 and the fact that (∆ (ii) and consequently (iii). 

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3.3.3. Nef cone and Kawamata-Viehweg vanishing, intersection numbers. The  nef cone of P(α, ξ) is given by the half line [0, ∞) in Ad−1 P(α, ξ) Q , the ample cone by its interior. P Definition P 3.3.15: We denote KC := − i∈C+ Di the minimal divisor and Anef := KC + (0, ∞) = (− i∈C+ Di , ∞).  P If we identify KC with its class i∈C+ αi in Ad−1 P(α, ξ) Q , we obtain that KC ≤ D for every P P D = i∈C ei Di for with −1 ≤ ei < 0 for all i ∈ C. In particular, KP(α,ξ) = − i∈C Di ≥ KC with equality if and only if C+ = C. This is the Kawamata-Viehweg theorem for toric 1-circuit varieties: Proposition 3.3.16: Let D := D ′ + E be a Weil divisor on P(α, ξ), where D′ is Q-ample and  P E = i∈C ei Di with −1 ≤ ei < 0 for all i ∈ C. Then H i P(α, ξ), O(D) = 0 for all i > 0.

Proof. The assertion follows from 3.3.13. Note that every integral divisor class which maps to Anef is already contained in FC. 

We point out that the converse of 3.3.16 is not true in general. As already illustrated in the introduction, there might be Weil divisors which do not map to a class in Anef and which have no cohomology. As the theorem is stated over Q, it only captures the offsets of the diophantine Frobenius problem. However, the theorem is sharp if L contains a Z-basis of N , in particular if P(α, ξ) is smooth. Also note that O(KC) by Proposition 3.3.13 has nonvanishing cohomology. P Definition 3.3.17: let D = i ci Di be a T -invariant divisor of P(α, ξ). Then we denote PD := {m ∈ MQ | li (m) ≥ −ci }.

Let D be an effective invariant Cartier divisor on P(α, ξ) and let τ ∈ ∆C(d−1) an inner wall. The intersection product D.V (τ ) can be read off the polyhedron PD . Namely, let PD,τ ⊂ PD the one-dimensional face of PD corresponding to τ . Then PD,τ is a bounded interval in MQ and it  is easy to see that the number of lattice points on PD,τ coincides with dim H 0 V (τ ), O(D)|V (τ ) = 1 + deg O(D)|V (τ ) . Let {i, j} = C \ τ (1). If we first restrict to the open subvariety U := Uσi ∪ Uσj of P(α, ξ), we obtain analogously to proposition 3.3.8 that Pic(U ) is generated by s −1 that PD,τ has lattice sτ αi · lcm{αi , αj } · Di , which corresponds to the class in Pic(U ) such  length 1. Over Q, we can identify Di with its class αi in Ad−1 P(α, ξ) Q ∼ = Q, and we obtain: sτ −1 α gcd{αi , αj }. s j Note that here, and for the rest of this work, we are only interested in intersections of curves with divisors. For simplicity, we will not distinguish between cycles and their degree. The inclusion U ֒→ P(α, ξ) does not change the intersection product, so above formula also holds on P(α, ξ). For any other k ∈ C, we then obtain by linearity that Dk .V (τ ) = ααki Di .V (τ ).  So we get a handy formula for the linear form on Ad−1 P(α, ξ) Q associated to τ : Di .V (τ ) =

Definition 3.3.18: We set

tC,τ := tα,ξ,τ :=

sτ gcd{αi , αj } sτ = lcm{αi , αj }−1 . s αi αj s

3.3.4. Fujita’s ampleness theorem for toric 1-circuit varieties. Recall that a Cartier divisor D is ample if ∆ coincides with the inner normal fan of PD . The divisor D is very ample if it is ample and moreover, for every σ ∈ ∆, the semigroup σM is generated by (PD ∩ M ) − mσ ,

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where mσ is the corner of PD corresponding to σ. We call a Q-divisor ample, if some integral multiple of it is ample. We want to give an ampleness criterion similar to proposition 3.3.16 in terms of intersection numbers. First we give an analog to a theorem of Fujino [Fuj03] (see also [Pay04]): P Proposition 3.3.19: Let E = i∈C ei Di , where −1 ≤ ei ≤ 0 for all i ∈ C and D an invariant Q-divisor on P(α, ξ). If for every inner wall τ of ∆C we have D.V (τ ) ≥ |C+ | − x, where x = 0 if P(α, ξ) is smooth and x = 1 otherwise, then D + E is nef. Proof. For every inner wall τ , we have D.V (τ ) = D.tC,τ = D · ssτ lcm{αi , αj }−1 ≥ |C+ | − x, where {i, j} = C \ τ (1). Thus D ≥ ssτ lcm{αi , αj } · (|C+ | − x). Hence D ≥ max{αi }i∈C+ · (|C+ | − x) ≥ KC, with equality if and only if P(α, ξ) is smooth.  We obtain some criteria for ampleness in the spirit of the toric Fujita ampleness theorem (see also [Mus02], [Pay04]): Corollary 3.3.20: Let P(α, ξ) and E, D be as before, such that D + E Cartier. If for every inner wall τ of ∆C (i) D.V (τ ) ≥ |C+ | + 1 and P(α, ξ) is smooth, then D + E is very ample; (ii) D.V (τ ) ≥ d + 1 and P(α, ξ) is not smooth, then D + E is very ample; (iii) D.V (τ ) ≥ |C+ | and P(α, ξ) is not smooth, then D + E is ample. Proof. For (i) note that all Di with i ∈PC+ are linearly equivalent and Di .V (τ ) = 1 for every inner wall τ and all i ∈ C+ . Thus D > i∈C+ Di = −KC and thus D + E is effective and nef, and therefore ample by smoothness of P(α, ξ). If P(α, ξ) is not minimal, we need sufficient conditions for D + E to be ample or very ample. To show that D + E is very ample, we have to verify that for every i ∈ C+ with associated maximal cone σi , the shifted polytope PD+E −mσi generates the semigroup σi,M , where mσi ∈ M is the corner of PD+E corresponding to σi . Let p1 , . . . , pd be the primitive vectors of the rays of the dual cone σ ˇi . Due to a criterion of Ewald and Wessels [EW91], it suffices to show that the simplex spanned by 0 and (d − 1) · pj is contained in PD+E − mσi . Note that the pj , where j ∈ C− , span a subcone of the recession cone of PD+E , i.e. N.pj ⊂ PD+E − mσi . So we have only to check the pj with j ∈ C+ . As we have seen before, the lattice distance of pj is given by s s −1 sτ αi lcm(αi , αj ). Let τi,j := σi ∩ σj , then we obtain D.V (τi,j ) ≥ sτ lcm(αi , αj ) · (d + 1) and thus D is at least the factor αi · (d + 1) larger than the minimal divisor Dij such that PDij ,τ has lattice length at least one. In fact, D ≥ max{αi · (d + 1) · Dij }j∈C+ \{i} . Thus we get that the lattice length of PD+E,τij ≥ d + 1 and (ii) follows. Assertion (iii) follows at once as from the estimate follows that pi ∈ PD+E − mσ for every i ∈ C.  3.3.5. Cohomology and resolutions of singularities. It is instructive to see the local cohomology vanishing in the context of classification of maximal Cohen-Macaulay modules. Assume that |C− | ∈ / {0, 1}, then the li span a strictly convex cone which gives rise to an affine toric variety Y . Recall that there is a natural map π : P(α, ξ) −→ Y which is a small resolution of singularities. Likewise, by flipping we obtain a second resolution π ′ : P(−α, ξ)) −→ Y . We π −1

(π ′ )−1

have two natural isomorphisms Ad−1 (Y ) −→ Ad−1 (P(α, ξ)) and Ad−1 (Y ) −→ Ad−1 (P(α, ξ)), which both are induced by the identity on Zn . These isomorphisms map any Weil divisor D on Y to its strict transforms π −1 D or (π ′ )−1 D, respectively, on P(α, ξ)) or P(−α, ξ)), respectively. Now, the question whether O(D) is a maximal Cohen-Macaulay sheaf can be decided directly on Y or, equivalently, on the resolutions:

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Theorem 3.3.21: Let Y be an affine toric variety whose associated cone σ is spanned by a circuit C and denote P(α, ξ) and P(−α, ξ) the two canonical small resolution of singularities. Then the sheaf O(D) is maximal Cohen-Macaulay if and only if Ri π∗ O(π −1 D) = Ri π∗′ O((π ′ )−1 D) = 0 for all i > 0. Proof. This toric variety corresponds to the toric subvariety of Y which is the complement of its unique fixed point, which we denote y. We have to show that Hyi Y, O(D) = 0 for all i < d. By Corollary 3.2.12, we have  Hyi Y, O(D) m = H i−2 (ˆ σy,m ; k)

for every m ∈ M , where σ ˆy denotes the simplicial model for the fan associated to Y \ {y}. Denote τ and τ ′ the cones corresponding to the minimal orbits of P(α, ξ) and P(−α, ξ), respecˆ C)V (τ ) = (∆ ˆ C)V (τ ′ ) both coincide with the subfan of σ generated tively. We observe that (∆ by its facets. It follows that the simplicial complexes relevant for computing the isotypical decomposition of Hyi Y, O(D) coincide with the simplicial complexes relevant for computing the   HVi P(α, ξ), O(π −1 D) and HVi ′ P(−α, ξ), O((π ′ )−1 D) , respectively, where V, V ′ denote the exceptional sets of the morphisms π and π ′ , respectively. By Proposition 3.3.14 the corresponding  cohomologies vanish for i < d iff D ∈ FC ∩ F−C. Now we observe that Γ Y, Riπ∗ O(π −1 D) = H i P(α, ξ), O(π −1 D) and Γ Y, Ri π∗′ O((π ′ )−1 D) = H i P(−α, ξ), O((π ′ )−1 D) . By Proposition 3.3.13, both cohomologies vanish for i > 0 iff D ∈ FC ∩ F−C.  Remark 3.3.22: The relation between maximal Cohen-Macaulay modules and the diophantine Frobenius problem has also been discussed in [Sta96]. See [Yos90] for a discussion of MCMfiniteness of toric 1-circuit varieties. Remark 3.3.23: The fiber of π over the exceptional locus again is a toric 1-circuit variety, a finite quotient of a weighted projective space. This variety is given by P(α+ , ξ + ), whose associated fan is contained in N/N C− . Here, ξ + : N/N C− −→ N/N C− is the morphism induced by ξ, and for α+ , the α+ i equal to αi divided by the greatest common divisor of all lcm{αj }j∈I , where I runs over all maximal proper subsets of C+ which contain i. 3.4. Discriminants and combinatorial aspects cohomology vanishing In this section we will concentrate on aspects of toric geometry which are related to its underlying linear algebra. A toric variety X is specified by the set of primitive vectors l1 , . . . , ln ∈ N and the fan ∆ supported on these vectors. We can separate three properties which govern the geometry of X and are relevant for cohomology vanishing problems: (i) the linear algebra given by the vectors l1 , . . . , ln and their linear dependencies as Q-vectors; (ii) arithmetic properties, which are also determined by the li , but considered as integral vectors; (iii) its combinatorics, which is given by the fan ∆. In the case of toric 1-circuit varieties it was possible to study the arithmetic aspects in isolation. For general toric varieties, this is no longer possible, because the three properties interact in much more complicated ways, which we have to keep track of. In this section we will describe how linear algebraic and combinatorial aspects are combined. In the sequel it will be convenient to consider the li as matrix. So we define: Definition 3.4.1: For any given set of vectors l1 , . . . , ln ∈ NQ we denote L the matrix whose rows are given by the li . For any subset I of [n] we denote LI the submatrix of L whose rows are given by the li with i ∈ I.

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We will also frequently make use of the following abuse of notion: Convention 3.4.2: We will usually identify subsets I ⊂ [n] with the corresponding subsets of {l1 , . . . , ln }. In particular, if C ⊂ [n] such that the set {li }i∈C forms a circuit, then we will also call C a circuit. Also, we will in general not distinguish between {li }i∈I and LI . Definition 3.4.3: We say that a fan ∆ is supported on l1 , . . . , ln if ∆(1) coincides with the set of rays generated by a subset of the li . Moreover, note that circuits are not required to span MQ . If some circuit C ⊂ [n] generates a subvector space of codimension r in NQ , then for some orientation C of C the variety X(∆C) is isomorphic to P(α, ξ) × (k∗ )r for some appropriate α and ξ. If C+ = C, the fan ∆−C is empty. By convention, in that case we define X(∆−C) := (k∗ )r as the associated toric variety. The relevant facts from section 3.3 can straightforwardly be adapted to this situation. Definition 3.4.4: We denote C(L) the set of circuits of L and C(L) the set of oriented circuits of L, i.e. the set of all orientations C, −C for C ∈ C(L). In subsection 3.4.1 we consider circuits of the matrix L and the induced stratification of Ad−1 (X)Q . In subsection 3.4.2 we will collect some well-known material on secondary fans from [GKZ94], [OP91], and [BFS90] and explain their relation to discriminantal arrangements. Subsection 3.4.3 then applies this to certain statements about the birational geometry of toric varieties and cohomology vanishing. 3.4.1. Circuits and discriminantal arrangements. Recall  that for any torus invariant P divisor D = i∈[n] ci Di , the isotypical components HVi X, O(D) m for some cohomology group ˆ I , where I = I(m) = {i ∈ [n] | li (m) < −ci }. So, the set of depend on simplicial complexes ∆ ˆ I depends on the chamber decomposition of MQ which is induced all possible subcomplexes ∆ by the hyperplane arrangement which is given by hyperplanes H1 , . . . , Hn , where c

Hi := {m ∈ MQ | li (m) = −ci }. The set of all relevant I ⊂ [n]} is determined by the map sc : MQ −→ 2[n] ,

m 7→ {i ∈ [n] | li (m) < −ci }.

Definition 3.4.5: For m ∈ MQ , we call sc the signature of m. We call the image of MQ in 2[n] the combinatorial type of c. Remark 3.4.6: The combinatorial type encodes what in combinatorics is known as oriented matroid (see [BLS+ 93]). We will not make use of this kind of structure, but we will find it sometimes convenient to borrow some notions. So, given l1 , . . . , ln , we would like to classify all possible combinatorial types, depending on c ∈ Qn . The natural parameter space for all hyperplane arrangements up to translation by some element m ∈ MQ is given by the set AQ ∼ = Qn /MQ , which is given by following short exact sequence: L D 0 −→ MQ −→ Qn −→ Ad−1 (X)Q = AQ −→ 0. Then the D1 , . . . , Dn are the images of the standard basis vectors of Qn . This procedure of constructing the Di from the li is often called Gale transformation, and the Di are the Gale duals of the li . c Now, a hyperplane arrangement Hi for some c ∈ Qn , is considered in general position if c the hyperplanes Hi intersect in the smallest possible dimension. When varying c and passing

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from one arrangement in general position to another with of a different combinatorial type, this necessarily implies that has to take place some specialization for some c ∈ Qn , i.e. the c corresponding hyperplanes Hi do not intersect in the smallest possible dimension. So we see that the combinatorial types of hyperplane arrangements with fixed L and varying induce a stratification of AQ , where the maximal strata correspond to hyperplane arrangements in general position. The determination of this stratification is the discriminant problem for hyperplane arrangements. To be more precise, let I ⊂ [n] and denote \ c Hi 6= 0}, HI := {c + MQ ∈ AQ | i∈I

i.e. HI represents the set of all hyperplane arrangements (up to translation) such that the hyperplanes {Hi }i∈I have nonempty intersection. The sets HI can be described straightforwardly by the following commutative exact diagram: (6)

HI _ 0

// MQ

MQ

L

// Qn

D

 // AQ

// 0

LI

 // QI

DI

 // AI,Q

// 0.

In particular, HI is a subvector space of AQ . Moreover, we immediately read off diagram (6): Lemma 3.4.7: (i) HI is generated by the Di with i ∈ [n] \ I. (ii) dim HI = n − |I| − dim(ker LI ). (iii) If J ⊆ I then HI ⊆ HJ . (iv) Let I, J ⊂ [n], then HI∪J ⊂ HI ∩ HJ . Note that in (iv), the reverse inclusion in general is not true. It follows that the hyperplanes among the HI are determined by the formula: |I| = rk LI + 1. By Lemma 3.4.7 (iii), we can always consider the minimal linearly dependent sets I, i.e. circuits, fulfilling this condition. It turns out that the hyperplane HC suffice to completely describe the discriminants of L: Lemma 3.4.8: Let I ⊂ [n], then HI =

\

HC ,

C⊂Icircuit

where, by convention, the right hand side equals AQ , if the li with i ∈ I are linearly independent. Hence, the stratification of AQ which we were looking for is completely determined by the hyperplanes HC . Definition 3.4.9: We denote the set {HC | C ⊂ [n] a circuit} the discriminantal arrangement of L. Remark 3.4.10: The discriminantal arrangement carries a natural matroid structure. This structure can be considered as another combinatorial invariant of L (or the toric variety X, respectively), its circuit geometry. Discriminantal arrangements seem to have been appeared

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first in [Cra84], where the notion of ’circuit geometry’ was coined. The notion of discriminantal arrangements stems from [MS89]. Otherwise, this subject seems to have been studied explicitly only in very few places, see for instance [Fal94], [BB97], [Ath99], [Rei99], [Coh86], [CV03], though it is at least implicit in the whole body of literature on secondary fans. Above references are mostly concerned with genericity properties of discriminantal arrangements. Unfortunately, in toric geometry, the most interesting cases (such as smooth projective toric varieties, for example) virtually never give rise to discriminantal arrangements in general position. Instead, we will focus on certain properties of nongeneric circuit geometries, though we will not undertake a thorough combinatorial study of these. Virtually all problems related to cohomology vanishing on a toric variety X must depend on the associated discriminantal arrangement and therefore on the circuits of L. In subsection 3.4.2 we will see that the discriminantal arrangement is tightly tied to the geometry of X. As we have seen in section 3.3, to every circuit C ⊂ [n] we can associate two oriented circuits. These correspond to the signature of the bounded chamber of the subarrangement in MQ given c by the Hi with i ∈ C (or better to the bounded chamber in MQ / ker LI , as we do no longer require that the li with i ∈ C span MQ ). Lifting this to AQ , this corresponds to the half spaces in AQ which are bounded by HC . Definition 3.4.11: Let C ⊂ [n] be a circuit, then we denote HC the half space in AQ bounded by HC corresponding to the orientation C. We obtain immediately: Lemma 3.4.12: Let C be a circuit of L and C an orientation of C. Then the hyperplane HC is separating, i.e. for every i ∈ [n] one of the following holds: (i) i ∈ [n] \ C iff Di ∈ HC ; (ii) if i ∈ C+ , then Di ∈ HC \ HC ; (iii) if i ∈ C− , then Di ∈ H−C \ HC . Now we are going to borrow some terminology from combinatorics. Consider any subvector space U of AQ which is the intersection of some of the HC . Then the set FU of all C ∈ C(L) such that HC contains U is called a flat. The subvector space is uniquely determined by the flat and vice versa. We can do the same for the actual strata rather than for subvector spaces. For this, we just need to consider instead the oriented circuits and their associated half spaces in AQ : any stratum S of the discriminantal arrangement uniquely determines a finite set FS of oriented circuits C such that S ⊂ HC. From the set FS we can reconstruct the closure of S: \ S= HC, C∈FS

We give a formal definition: Definition 3.4.13: For any subvector space U ⊂ AQ which is a union of strata of the discriminantal arrangement, we denote FU := {C ∈ C(L) | U ⊂ HC } the associated flat. For any single stratum S ⊂ AQ of the discriminantal arrangement, we denote FS := {C ∈ C(L) | U ⊂ HC} the associated oriented flat. The notion of flats gives us some flexibility in handling strata. Note that flats reverse inclusions, i.e. S ⊂ T iff FT ⊂ FS . Moreover, if a stratum S is contained in some HC , then its oriented flat contains both HC and H−C, and vice versa. So from the oriented flat we can reconstruct FC and thus the subvector space of AQ generated by S.

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Definition 3.4.14: Let S := {S1 , . . . , Sk } be a collection of strata of the discriminantal arrangement. We call k \ F Si FS := i=1

the discriminantal hull of S.

T

The discriminantal hull defines a closed cone in AQ which is given by the intersection C∈FS HC. This cone contains the union of the closures S i , but is bigger in general.

Lemma 3.4.15: (i) Let S = {S1 , . . . , Sk } be a collection of discriminantal strata whose union T is a closed cone in AQ . then FS = ki=1 FSi . (ii) Let S = {S1 , . . . , Sk } be a collection of discriminantal strata and U the subvector space of AQ generated by the Si . Then the forgetful map FS → FU is surjective iff FS = FSi for some i. S Proof. For (i) just note that because ki=1 Sk is a closed cone, it must be an intersection T of some HC. For (ii): the set C∈FS HC is a cone which contains the convex hull of all the S i . If some C is not in the image of the forgetful map, then the hyperplane HC must intersect the relative interior of this cone. So the assertion follows.  c

3.4.2. Secondary Fans. For any c ∈ Qn the arrangement Hi induces a chamber decomposition of MQ , where the closures of the chambers are given by PcI := {m ∈ MQ | li (m) ≤ −ci for i ∈ I and li (m) ≥ −ci for i ∈ / I} for every I ⊂ [n] which belongs to the combinatorial type of c. In particular, c represents an element D ∈ AQ with \ D∈ CI , I∈sc (MQ )

where CI is the cone in AQ which is generated / I for P by the −Di for i ∈ I and theI Di with i ∈ some I ⊂ [n]. For an invariant divisor D = i∈[n] ci Di we will also write PD instead of PcI . If I = ∅, we will occasionally omit the index I. The faces of the CI can be read off directly from the signature: Proposition 3.4.16: Let I ⊂ [n], then CI is an nonredundant intersection of the HC with C− ⊂ I and C+ ∩ I = ∅. Proof. First of all, it is clear that CI coincides with the intersection of half spaces \ HC. CI = C+ ⊂I C− ∩I=∅

For any HC in the intersection let HC its boundary. Then HC contains a cone of codimension 1 in AQ which is spanned by Di with i ∈ [n] \ (C ∪ I) and by −Di with i ∈ I \ C which thus forms a proper facet of CI .  Recall that the secondary fan of L is a fan in AQ whose maximal cones are in one-to-one correspondence with the regular simplicial fans which are supported on the li . That is, if c is chosen sufficiently general, then the polyhedron Pc∅ is simplicial and its inner normal fan is a simplicial fan which is supported on the li . Wall crossing in the secondary fan then corresponds to a transition ∆C −→ ∆−C as in section 3.3. Clearly, the secondary fan is a substructure of the

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discriminantal arrangement in the sense that its cones are unions of strata of the discriminantal arrangements. However, the secondary fan in general is much coarser than the discriminantal arrangement, as it only keeps track of the particular chamber Pc∅ . In particular, the secondary fan is only supported on C∅ which in general does not coincide with AQ . Of course, there is no reason to consider only one particular type of chamber — we can consider secondary fans for every I ⊂ [n] and every type of chamber PcI . For this, observe first that, if B is a subset of [n] such that the li with i ∈ B form a basis of MQ , then the complementary Gale duals {Di }i∈B / form a basis of AQ . Then we set: Definition 3.4.17: Let I ⊂ [n] and B ⊂ [n] such that the li with i ∈ B form a basis of MQ , then we denote KBI the cone in AQ which is generated by −Di for i ∈ I \ B and by Di for i ∈ [n] \ (I ∪ B). The secondary fan SF(L, I) of L with respect to I is the fan whose cones are the intersections of the KBI , where B runs over all bases of L. Note that SF(L, ∅) is just the secondary fan as usually defined. Clearly, the chamber structure of the discriminantal arrangement still refines the chamber structure induced by all secondary arrangements. But now we have sufficient data to even get equality: Proposition 3.4.18: The following induce identical chamber decompositions of AQ : (i) the discriminantal arrangement, (ii) the intersection of all secondary fans SF(L, I), (iii) the intersection of the CI for all I ⊂ [n]. Proof. Clearly, the facets of every orthant CI span a hyperplane which is part of the discriminantal arrangement, so the chamber decomposition induced by the secondary fan is a refinement of the intersection of the CI ’s. The CI induce a refinement of the secondary fans as follows. Without loss of generality, it suffices to show that every KB∅ is the intersection of some CI . We have \ CI . KB∅ ⊆ I⊂B

KB∅ ,

On the other hand, for every facet of we choose I such that CI shares this face and KB∅ is contained in CI . This can always be achieved by choosing I so that every generator of CI is in the same half space as KB∅ . The intersection of these CI then is contained in KB∅ . Now it remains to show that the intersection of the secondary fans refines the discriminantal arrangement. This actually follows from the fact, that for every hyperplane HC , one can choose a minimal generating set which we can complete to a basis of AQ from the Di , where i ∈ / C. By varying the signs of this generating set, we always get a simplicial cone whose generators are contained in some secondary fan, and this way HC is covered by a set of facets of secondary cones.  The maximal cones in the secondary fan SF(L, ∅) correspond to regular simplicial fans supported on T l1 , .∅. . , ln . More precisely, if ∆ denotes such a fan, then the corresponding cone is given by B KB , where B runs over all bases among the li which span a maximal cone in ∆. This definition makes sense for any fan ∆ supported on the li , we can single out a specific cone ˆ for ∆, we set: in SF(L, ∅). Choosing a simplicial model ∆ Definition 3.4.19: Let ∆ be a fan supported on L, then we set: \ nef(∆) := KB∅ ˆ B∈∆ B basis in L

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and denote Fnef = Fnef(∆) the discriminantal hull of nef(∆). ˆ with the set of corresponding primitive Note that by our conventions we identify B ∈ ∆ vectors, or the corresponding rows of L, respectively. Of course, nef(∆) is just the nef cone of the toric variety associated to ∆. Proposition 3.4.20: We have: nef(∆) =

\

HC.

ˆ ∆∩(∆ C )max 6=∅

Proof. For some basis B ⊂ [n], the cone KB∅ is simplicial, and for every i ∈ [n] \ B, the facet of KB∅ which is spanned by the Dj with j ∈ / B ∪ {i}, spans a hyperplane HC in P . This hyperplane corresponds to the unique circuit C ⊂ B ∪ {i}. As we have seen before, a maximal cone in ∆C is of the form C \ {j} for some j ∈ C+ . So we have immediately: \ KB = HC ∃F ∈(∆C )max with F ⊂B

and the assertion follows.



ˆ is a regular simplicial fan, then nef(∆) is a maximal cone in the Remark 3.4.21: If ∆ = ∆ secondary fan. Let C be an oriented circuit such that ∆ is supported on ∆C in the sense of [GKZ94], §7, Def. 2.9, and denote ∆′ the fan resulting in the bistellar operation by changing ∆C to ∆−C. Then, by [GKZ94], §7, Thm. 2.10, the hyperplane HC is a proper wall of nef(∆) iff ∆′ is regular, too. 3.4.3. Birational toric geometry, rational divisors, and vanishing theorems. Circuits and their related numerical properties are an important tool in toric geometry, in particular in the context of the toric minimal model program (see [Rei83] and [Mat02b], Chapter 14) and the classification of smooth toric varieties (see [Oda88], §1.6, for instance). The purpose of this subsection is to clarify the relation of some standard constructions with the intrinsic circuit geometry of a toric variety. Moreover, we will give new proof of some standard vanishing theorems from this point of view. In this section ∆ denotes a fan associated to a toric variety X. L denotes the row matrix of primitive vectors of rays in ∆. We always assume that the support of ∆ in NQ coincides with the positive span of the li . Note that this in particular implies that Pic(X) is torsion free and N 1 (X) = Pic(X). Some remarks on Q-Cartier divisors on toric varieties. Recall that a Q-divisor on X is Q-Cartier P if an integral multiple is Cartier in the usual sense. A torus invariant Weil divisor D = i∈[n] ci Di is Q-Cartier iff for every σ ∈ ∆ there exists some mσ ∈ MQ such that ci = li (m) for all i ∈ σ(1). Proposition 3.4.22: Let D ∈ Ad−1 (X) be a Weil divisor, which is Q-Cartier. Then O(D) is maximally Cohen-Macaulay. Proof. The MCM property is a local property. So, without loss of generality, it suffices to consider the restriction of D to some Uσ . Because D is Q-Cartier, the hyperplane arrangement 0 c Hi coincides with Hi up to a translation by mσ . Therefore sc (MQ ) coincides with s0 (MQ ). 0 Also, all strata of the hyperplane arrangement Hi , except possibly the trivial stratum, are unbounded, and thus contain lattice points. So, because the structure sheaf is MCM, it follows  that all simplicial complexes σ ˆI are acyclic for [n] 6= I ∈ s0 = sc , hence O(D) is MCM.

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Remark 3.4.23: See also [BG03] for another proof of Proposition 3.4.22. The MCM-property is useful, as it allows to replace the Ext-groups by cohomologies in Serre duality: Proposition 3.4.24: Let X be a normal variety with dualizing sheaf ωX and F a coherent sheaf on X such that for every x ∈ X, the stalk Fx is MCM over OX,x . Then for every i ∈ Z there exists an isomorphism   ExtiX F, ωX ∼ = H i X, Hom(F, ωX ) . Proof. For any two OX -modules F, G there exists the following spectral sequence  E2pq = H p X, ExtqOX (F, G) ⇒ Extp+q OX (F, G).

We apply this spectral sequence to the case G = ωX . For every closed point x ∈ X we have the following identity of stalks: q ExtqOX (F, ωX )x ∼ = ExtOX,x (Fx , ωX,x ).

As F is maximal Cohen-Macaulay, the latter vanishes for all q > 0, and thus the sheaf ExtqOX (F, ωX ) is the zero sheaf for all q > 0. So the above spectral sequence degenerates and we obtain an isomorphism p H p (X, Hom(F, ωX )) ∼ = Ext (F, ωX ) X

for every p ∈ Z.



P In the case where X a toric variety, we have ωX ∼ = O(KX ), where KX = − i∈[n] Di . Then, if F = O(D) for some D ∈ A, we can identify Hom(O(D), ωX ) with O(KX − D): Corollary 3.4.25: Let X be a toric variety and D a Weil divisor such that O(D) is an MCM sheaf. Then there is an isomorphism:   ExtiX O(D), ωX ∼ = H i X, OX (KX − D) . And by Grothendieck-Serre duality:

Corollary 3.4.26: If X is a complete toric variety and D a Weil divisor such that O(D) is an MCM sheaf, then   H i X, O(D) ∼ = H d−i X, O(KX − D) ˇ.

The Picard group. The Picard group in a natural way coincides with a flat of the discriminantal arrangement: Theorem 3.4.27 (see [Eik92], Theorem 3.2): Let X be any toric variety, then: \ \ HC . Hσ(1) = Pic(X)Q = σ∈∆max

C∈C(Lσ(1) ) σ∈∆max

Proof. As remarked before, a Q-Cartier divisor is specified by a collection {mσ }σ∈∆ ⊂ MQ . c In particular, all for every σ ∈ ∆, the hyperplanes Hi with i ∈ σ(1) have nonempty intersection. So the first equality follows. The second equality follows from Lemma 3.4.8. 

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Nef and Mori cone. The nef cone nef(X) is nothing but nef(∆) as defined in subsection ˆ 6= ∅. Moreover, in the 3.4.2. Let us denote F′nef ⊂ Fnef the subset of those C such that ∆C ∩ ∆ case that ∆ is simplicial, denote F′′nef ⊂ Fnef the subset such that ∆ is supported on ∆C in the sense of [GKZ94], §7, Def. 2. Then the following is a consequence of Proposition 3.4.20: Theorem 3.4.28: The nef cone of X is given by the following intersection in AQ : \ \ HC. HC = nef(X) = C∈Fnef

C∈F′nef

ˆ be a simplicial fan supported on L. Then every inner facet τ ∈ ∆(d − 1), Let ∆ = ∆ has a canonically associated circuit. Namely, τ is contained in precisely two maximal cones σ, σ ′ ∈ ∆(d), and the set σ(1) ∪ σ ′ (1) contains precisely d + 1 elements, and σ(1) as well as σ ′ (1) form a basis of NQ . Therefore, the set σ(1) ∪ σ ′ (1) contains a unique circuit. Definition 3.4.29: Let τ be an inner facet of a simplicial fan ∆. Then we denote C(τ ) its canonically associated circuit. Any fan of the form ∆C for some oriented circuit C is simplicial. We can make use of the calculations of section 3.3 to define linear forms on Pic(X)Q . Let C ∈ C(L) be any oriented circuit, then by the short exact sequence 0 −→ HC −→ Ad−1 (X)Q −→ AC,Q −→ 0, we can lift the linear forms tC,τ , where τ an inner wall of ∆C, to linear forms on Ad−1 (X)Q : ¯ its image in AC,Q and set tC,τ (D) := tC,τ (D). ¯ It follows for every D ∈ Ad−1 (X)Q we denote D lcm{α ,α } easily that any tC,τ and tC,τ ′ are proportional to each other by the factors ssτ′ lcm{αpi ,αjq } , where τ P {i, j} = C \ τ (1) and {p, q} = C \ τ ′ (1) and the defining relation for C is i∈C αi li = 0.

Remark 3.4.30: Note that in the case that X is nonsimplicial, the form tC,τ should not be identified with an actual curve class. We only have a linear form on Ad−1 (X)Q . We can think of the tC,τ as ’virtual’ curve classes on X. Also note that the facet τ in general is not realized as a facet in ∆. If X is complete, by the nondegenerate pairing N1 (X)Q ⊗Q Pic(X)Q −→ Q we can identify N1 (X)Q with a quotient vector space of the dual vector space Ad−1 (X)ˇQ and in fact the tC,τ become rational equivalence classes of curves after projection to N1 (X)Q . Denote these projections t¯C,τ , then we have by the duality between the nef cone and the Mori cone: Theorem 3.4.31: Let X be a complete toric variety, then the Mori cone of X in N1 (X)Q is generated by the t¯C,τ , where C ∈ F′nef . Remark 3.4.32: Classes of circuits on which the fan ∆ is supported have been considered earlier in [Cas03] and were called “contractible classes”. A contractible class is extremal iff the result of the associated flip is a projective toric variety again. See also [Bon00] for examples. If X is simplicial and projective, the Mori cone is a strictly convex polyhedral cone in N1 (X) = Ad−1 (X)ˇQ and by the theorem, F′nef contains the set of its extremal rays. The Mori cone is generated by the tC,τ such that V (τ ) is an extremal curve. However, in general, F′nef is strictly bigger than the set of extremal curve classes.

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The Iitaka dimension of a nef divisor and Kawamata-Viehweg vanishing. Let D be a Cartier divisor on some normal variety X, and denote N (X, D) := {k ∈ N | H 0 X, O(kD) 6= 0}. Then the Iitaka dimension of D is defined as κ(D) :=

max {dim φk (X)},

k∈N (X,D)

where φk : X _ _ _// P|kD| is the family of morphisms given by the linear series |kD|. P In the case where X is a toric variety and D = i∈[n] ci Di invariant, the Iitaka dimension of D is just the dimension of PkD for k >> 0. For a Q-Cartier divisor D, we define its Iitaka dimension by κ(D) := κ(rD) for r > 0 such that rD is Cartier. If D is a nef divisor, then the morphism φ : X −→ P|D| is torus equivariant, its image is a projective toric variety of dimension κ(D) whose associated fan is the inner normal fan of PD . If κ(D) < d, then necessarily D is contained in some hyperplane HC such that C+ = C for some orientation C of C. The toric variety associated to C is isomorphic to a finite cover of a weighted projective space. This kind of circuit will play an important role later on, so that we will give it a distinguished name: Definition 3.4.33: We call a circuit C such that C = C+ for one of its orientations, fibrational. By Proposition 3.4.16, this implies that D is contained in the intersection of nef(X) with the effective cone of X, which we identify with C∅ . More precisely, it follows from linear algebra that D is contained in all HC where C is fibrational and li (PD ) = 0 for all i ∈ C. Definition 3.4.34: Let D ∈ Ad−1 (X)Q , then we denote fib(D) ⊂ C(L) the set of fibrational circuits such that D ∈ HC . The fibrational circuits of a nef divisor D tell us immediately about its Iitaka dimension: Proposition 3.4.35: Let D be a nef Q-Cartier divisor. Then κ(D) = d − rk LT , where T := S C. C∈fib(D)

Proof. We just remark that rk LT is the dimension of the subvector space of MQ which is generated by the li which are contained in a fibrational circuit. 

vanishing is Recall from section 3.3.2 that for a toric 1-circuit variety P(α, ξ), cohomology   Q determined by the set FC ⊂ Ad−1 P(α, ξ) . The image of FC in Ad−1 P(α, ξ) Q ∼ = contains all P classes D > KC, where KC = − i∈C+ Di (see section 3.3.3), i.e. all classes which are contained in the open interval (KC, ∞). In particular, in the case where C is not fibrational, it also contains P the canonical divisor KP(α,ξ) = − i∈C Di .  Proposition 3.4.36: Let X be a complete toric variety and D a nef divisor, then H i X, O(−D) = 0 for i 6= κ(D). c

Proof. Consider the hyperplane arrangement given by the Hi in MQ . Let m ∈ MQ and ˆ I can be characterized as follows. Consider Q ⊂ PD I = sc (m). Then the simplicial complex ∆ c the union of the set faces of PD which are contained in any Hi with i ∈ I. This is precisely the portion of PD , which the the point m “sees”, and therefore contractible, where the convex hull of ˆ I is contractible with an Q and m provides the homotopy between Q and m. Therefore, every ∆ ˆ exception for I = ∅, because ∆∅ = ∅, which is not acyclic with respect to reduced cohomology. −c Now we pass to the inverse, i.e. we consider the signature of −m with respect to Hi . Then [n] for any such −m which does not sit in the relative interior of the polytope P−c , there exists

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ˆ J is contractible and s−c (m) = [n] \ J. As ∆ ˆ is m′ ∈ MQ with signature sc (m′ ) =: J such that ∆ homotopic to a d − 1-sphere, we can apply Alexander duality and thus the simplicial complex ˆ [n]\J is acyclic. Thus there remain only the elements in the relative interior of P [n] . Let m ∆ −c ˆ be such an element with signature I, then ∆I is isomorphic to a d − κ(D) − 1-sphere, and the assertion follows.  This proposition implies the toric Kodaira and Kawamata-Viehweg vanishing theorems (see also [Mus02]): Theorem 3.4.37 (Kodaira & Kawamata-Viehweg): Let X be a complete toric variety and D, P E Q-divisors with D nef and E = i∈[n] ei Di with −1 < ei < 0 for all i ∈ [n]. Then:  (i) if D is integral, then H i X, O(D + KX ) = 0 for all i 6= 0, d − κ(D); (ii) if D + E is a Weil divisor, then H i X, O(D + E) = 0 for all i > 0.

Proof. By Proposition 3.4.22 wecan apply Serre duality (Corollary 3.4.26) and obtain  H i X, O(D + KX ) ∼ = H d−i X, O(−D and (i) follows from Proposition 3.4.36. For (ii): D + E is contained the interior of every half space KX + HC for C ∈ Fnef , and the result follows.  3.5. Arithmetic aspects of cohomology vanishing

In this section we are going to combine aspects from sections 3.3 and 3.4. In particular, we want to derive vanishing results for integral divisors which cannot directly be derived from the setting of Q-divisors. P From now on we assume that the li are integral. Recall that for any integral divisor D = i∈[n] ci Di and any torus invariant closed subvariety V of X, vanishing of  HVi X, O(D) depends on two things: (i) whether the set of signatures sc (MQ ) consists of I ⊂ [n] such that the relative cohomology ˆ I, ∆ ˆ V,I ; k) vanish, and, groups H i−1 (∆ i−1 ˆ ˆ (ii) if H (∆I , ∆V,I ; k) for one such I, whether the corresponding polytope PcI contain lattice points m with sc (m) = I. In the Gale dual picture, the signature sc (MQ ) coincides with the set of I ⊂ [n] such that the class of D in Ad−1 (X)Q is contained in CI . For fixed I, the classes of divisors D in Ad−1 (X) such that the equation li (m) < −ci for i ∈ I and li (m) ≥ −ci for i ∈ / I is satisfied, is counted by the generalized partition function. That is, by the function X X ki Di − ki Di = D where ki > 0 for i ∈ I} . D 7→ {(k1 , . . . , kn ) ∈ Nn | i∈[n]\I

i∈I

So, in the most general picture, we are looking for D lying in the common zero set of the vector partition function for all relevant signatures I of D. In general, this is a difficult problem to determine these zero sets, and it is hardly necessary for practical purposes.

Remark 3.5.1: The diophantine Frobenius problem (also known as money change problem or denumerant problem) is a classical problem of number theory and so far it is unsolved, though for fixed n, there exist polynomial time algorithms to determine the zeros of the vector partition function. Though there do exist explicit formulas for the Ehrhart quasipolynomials (see for instance [CT00]), a general closed solution is only known for the case n = 2. For a general overview we refer to the book [Ram05]. Vector partition functions play an important role in the combinatorial theory of rational polytopes and have been considered, e.g. in [Stu95], [BV97] (see also references therein). In [BV97] closed expressions in terms of residue formulas

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have been obtained. Moreover it was shown that the vector partition function is a piecewise quasipolynomial function, where the domains of quasipolynomiality are chambers (or possibly unions of chambers) of the secondary fan. In particular, for if Pc∅ is a rational bounded poly∅ topy, then the values of the vector partition function for Pk·c for k ≥ 0, is just the Ehrhart quasipolynomial. A first — trivial — approximation is given by the observation that the divisors D where the vector P partition function takes a nontrivial value map to the cone CI , shifted by the offset eI := − i∈I ei . This offset is essentially the same as the offset KC of section 3.3.

Definition 3.5.2: We denote O′ (L, I) the saturation of cone generated the −Di for i ∈ I and the Di for i ∈ / I and O(L, I) := eI + O′ (L, I). Moreover, we denote Ω(L, I) the zero set in O(L, I) of the vector partition function as defined above.

In the next step we want to approximate the sets Ω(L, I) by reducing to the classical diophantine Frobenius problem. For this, fix some I ⊂ [n] and consider some polytope PcI . It follows from Proposition 3.4.16 that D is contained in the intersection of half spaces HC for C ∈ C(L) such that C− = C ∩ I. In the polytope picture, we can interpret this as follows. For every C and its underlying circuit C, we set PcC := {m ∈ MQ | li (m) ≤ −ci for i ∈ C− and li (m) ≥ −ci for i ∈ C+ }. Consequently, we get PcI =

\

PcC,

C

where the intersection runs over all C ∈ C(L) with C− = C ∩ I. It follows that if there exists a compatible oriented circuit C such that PcC does not contain a lattice point, then PcI also does not contain a lattice point. We want to capture this by considering an arithmetic analogue of the discriminantal arrangement in Ad−1 (X) rather than in Ad−1 (X)Q . Using the integral pendant to diagram (6) (compare definition 3.3.12): Definition 3.5.3: Consider the morphism ηI : Ad−1 (X) ։ AI . Then we denote ZI its kernel. For I = C and C some orientation of C we denotePby FC the preimage in Ad−1 (X) of the P complement of the semigroup consisting of elements i∈C− ci Di − i∈C+ ci Di , where ci ≥ 0 for i ∈ C− and ci > 0 for i ∈ C+ . We set FC := FC ∩ F−C.

So, there are two candidates for a discriminantal arrangement in Ad−1 (X), the ZC on the one hand, and the FC on the other. Definition 3.5.4: We denote: • {ZC }C∈C(L) the integral discriminantal arrangement, and • {FC }C∈C(L) the Frobenius discriminantal arrangement.

The integral discriminantal arrangement has similar properties as the HI , as they give a solution to the integral discriminant problem (compare Lemma 3.4.8): Lemma 3.5.5: Let I ⊂ [n], then ZI =

\

ZC .

C∈C(LI )

As in the rational case, we can use this to locate the integral Picard group in Ad−1 (X):

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Theorem 3.5.6 (see [Eik92], Theorem 3.2): Let X be any toric variety, then: \ \ ZC . Zσ(1) = Pic(X) = σ∈∆max

C∈C(Lσ(1) ) σ∈∆max

Proof. A Cartier divisor is specified by a collection {mσ }σ∈∆ ⊂ M such that the hyperplanes HiC with i ∈ σ(1) intersect in integral points. So the first equality follows. The second equality follows from Lemma 3.5.5.  The Frobenius discriminantal arrangement is not as straightforward. First, we note the following properties: Lemma 3.5.7: Let C ∈ C(L), then: (i) FC is nonempty; (ii) the saturation of ZC in Ad−1 (X) is contained in FC iff C is not fibrational. Proof. The first assertion follows because FC contains all elements which map to the open interval (KC, K−C) in AC,Q . For the second assertion, note that the set {m ∈ M | li (m) = 0 for all i ∈ C} is in FC iff C+ 6= C for either orientation C of C.  Lemma 3.5.7 shows that the FC are thickenings of the ZC with one notable exception, where C is fibrational. In this case, FC can be considered as parallel to, but slightly shifted away from ZC . In the sequel we will not make any explicit use of the ZC anymore, but these facts should be kept in mind. Regarding the Frobenius discriminantal arrangement, we want also to consider integral versions of the discriminantal strata: Definition 3.5.8: Let C ∈ C(L) and let FS be a discriminantal hull of S = {S1 , . . . , Sk }, then we denote \ AS := FC. C∈FS

the arithmetic core of FS . In the special case FS = Fnef we write Anef .

Remark 3.5.9: The notion core refers to the fact that we consider all FC, instead of a nonredundant subset describing the set S as a convex cone. We will use arithmetic cores to derive arithmetic versions of known vanishing theorems formulated in the setting of Q-divisors and to get refined conditions on cohomology vanishing. This principle is reflected in the following theorem: Theorem 3.5.10: Let V be a T -invariant closed subscheme of X and S a discriminantal  i X, O(D) = 0 for some i and for all integral divisors D ∈ S, then stratum in Ad−1 (X) . If H Q V  also HVi X, O(D) = 0 for all D ∈ AS .

Proof. Without loss of generality we can assume that dim S > 0. Consider some nonempty I for some I ⊂ [n]. Then for any such I, we can choose some multiple of kD such that Pkc   contains a lattice point. But if HVi X, O(D) = 0, then also HVi X, O(kD) = 0, hence ˆ I, ∆ ˆ V,I ; k) = 0. Now, any divisor D ′ ∈ AS which does not map to S, is contained in H i−1 (∆ FC for all C ∈ FS and therefore for any I which is in the signature for D′ but not for D, the equations li (m) < −c′i for i ∈ I and li (m) ≥ −c′i for i ∈ / I cannot have any integral solution.  PcI

We apply Theorem 3.5.10 to Anef :

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Theorem 3.5.11 (Arithmetic version of Kawamata-Viehweg vanishing): Let X be a complete  toric variety. Then H i X, O(D) = 0 for all i > 0 and all D ∈ Anef .

Proof. We know that the assertion is true if D is nef. Therefore we can apply Theorem 3.5.10 to the maximal strata S1 , . . . , Sk of nef(X). Therefore the assertion is true for D ∈ T A k Si . To prove the theorem, we have to get rid of the FC, where HC intersects the relative i=1 interior of a face of nef(X). Let C be such a circuit and R the face. Without loss of generality, dim R > 0. Then we can choose elements D ′ in R at an arbitrary distance from HC, i.e. such that the polytope PcC becomes arbitrarily big and finally contains a lattice point. Now, if we move outside nef(X), but stay inside Anef , the lattice points of PcC cannot acquire any cohomology and the assertion follows.  One can imagine an analog of the set AS in Ad−1 (X)Q as the intersection of shifted half spaces \ KC + HC. C∈FS

The main difference here is that one would picture the proper facets of this convex polyhedral set as “smooth”, whereas the proper “walls” of AS have “ripples”, which arise both from the fact that the groups AC may have torsion, and that we use Frobenius conditions to determine the augmentations of our half spaces. In general, the set FS is highly redundant, when it comes to determine S, which implies that above intersection does not yield a cone but rather a polyhedron, whose recession cone corresponds to S. In the integral situation we do not quite have a recession cone, but a similar property holds: Proposition 3.5.12: Let V ⊂ X be a closed invariant subscheme and S  = {S1 , . . . , Sk } a collection of discriminantal stata different from zero such that HVi X, O(D) = 0 for D ∈ AS . Then for any nonzero face of its discriminantal hull S there exists the class of an integral divisor D′ ∈ S such that the intersection of the half line D + rD ′ for 0 ≤ r ∈ Q with AS contains infinitely many classes of integral divisors. Proof. T Let R ⊂ S be any face of S, then the vector space spanned by R is given by an intersection C with C∈K HC for T a certain subset K ⊂ FS . We assume that K is maximal with this property. The intersection C∈K FC is invariant with respect to translations along certain (though not necessarily all) T D′ ∈ R. This implies that the line (or any half line, respectively), ′ generated by D intersects C∈K FC in infinitely many points. As K is maximal, there is no other C ∈ FC parallel to R and the assertion follows.  The property of Proposition 3.5.12 is necessary for elements in AS , but not sufficient. This leads to the following definition: Definition 3.5.13: Let S = {S1 , . . . , Sk } be a collection of nonzero discriminantal strata and D ∈ Ad−1 (X) such that the property of Proposition 3.5.12 holds. If S is not contained in AS , then we call D AS -residual. If S = 0, then we write 0-residual instead of A0 -residual. T Note that, by definition, every divisor outside C∈C(L) FC is 0-residual. In the next subsections we will consider several special cases of interest for cohomology vanishing, which are not directly related to Kawamata-Viehweg vanishing theorems. In subsection 3.5.1 we will consider global cohomology vanishing for divisors in the inverse nef cone. In subsection 3.5.2 we will present a more explicit determination of this type of cohomology

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vanishing for toric surfaces. Finally, in subsection 3.5.3, we will give a geometric criterion for determing maximally Cohen-Macaulay modules. 3.5.1. Nonstandard Cohomology Vanishing. In this subsection we want to give a qualitative description of cohomology vanishing which is related to divisors which are inverse to nef divisors of Iitaka dimension 0 < κ(D) < d. We show the following theorem:  Theorem 3.5.14: Let X be a complete d-dimensional toric variety. Then H i X, O(D) = 0 for every i and all D which are contained in some A−F , where F is a face of nef(X) which cointains nef divisors of Iitaka dimension 0 < κ(D) < d. If A−F is nonempty, then it contains infinitely many divisor classes. T Proof. Recall that such a divisor, as a Q-divisor, is contained in the intersection C∈fib(D) HC and therefore it is in the intersection of the nef cone with the boundary of the effective cone  of i X, O(D ′ ) = 0 X by Proposition 3.4.16. Denote this intersection by F . Then we claim that H  for all D′ ∈ A−F . By Corollary 3.4.36 we know that H i X, O(E) = 0 for 0 ≤ i < d for any divisor E in the interior of the inverse nef cone. This implies that H i X, O(E) = 0 for any  E ∈ A−nef and hence H i X, O(D ′ ) = 0 for any D′ ∈ A−F , because A−F ⊂ A−nef . The latter assertion follows from the fact that the assumption on the Iitaka dimension implies that the face F has positive dimension.  Note that criterion is not very strong, as it is not clear in general whether the set A−F is nonempty. However, this is the case in a few interesting cases, in particular for toric surfaces, as we will see in the next subsection. The following remark shows that our condition indeed is rather weak in general: Remark 3.5.15: The inverse of any big and nef divisor D with the property  that PD does not i contain any lattice point in its interior has the property that H X, O(D) = 0 for all i. This follows directly from the standard fact in toric geometry that the Euler characteristics χ(−D) counts the inner lattice points of the lattice polytope PD . 3.5.2. The case of complete toric surfaces. Let X be a complete toric surface. We assume that the li are circularly ordered. We consider the integers [n] as system of representatives for Z/nZ, i.e. for some i ∈ [n] and k ∈ Z, the sum i + k denotes the unique element in [n] modulo n. Proposition 3.5.16: Let X be a complete toric surface. Then nef(X) = S, where S is a single stratum of maximal dimension of the discriminantal arrangement. Proof. X is simplicial and projective and therefore nef(X) is a cone of maximal dimension in A1 (X)Q . We show that no hyperplane HC intersects the interior of nef(X). By Proposition 3.4.16 we can at once exclude fibrational circuits. This leaves us with non-fibrational circuits C with cardinality three, having orientation C with |C+ | = 2. Assume that D is contained in the interior of H−C. Then there exists m ∈ MQ such that C+ ⊂ sD (m), which implies that the c hyperplane Hi for {i} = C− does not intersect PD , and thus D cannot be nef. It follows that nef(X) ⊂ HC.  Now assume there exist p, q ∈ [n] such that lq = −lp , i.e. lp and lq represent a onedimensional fibrational circuit of L. Then for any nef divisors D which is contained in Hp,q , the associated polytope PD is one-dimensional. The only possible variation for PD is its length in terms of lattice units. So we can conclude that nef(X) ∩ Hp,q is a one-dimensional face of nef(X).

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Definition 3.5.17: Let X be a complete toric surface and C = {p, q} such that lp = −lq . Then we denote Sp,q the relative interior of −nef(X) ∩ HC . Moreover, we denote Ap,q the arithmetic core of Sp,q . Our aim in this subsection is to prove the following: Theorem 3.5.18: Let  X be a complete toric surface. Then there are only S finitely many divisors i D with H X, O(D) = 0 for all i > 0 which are not contained in Anef ∪ Ap,q , where the union ranges over all pairs {p, q} such that lp = −lq .

We will prove this theorem in several steps. First we show that the interiors of the CI such ˆ I ; k) 6= 0 cover all of A1 (X)Q except nef(X) and −nef(X). that H 0 (∆ P Proposition 3.5.19: Let D = i∈[n] ci Di be a Weil divisor which is not contained in nef(X) c or −nef(X), then the corresponding arrangement Hi in MQ has a two-dimensional chamber PcI ˆ I has at least two components. such that complex ∆ T Proof. Recall that nef(X) = HC, where the intersection runs over all oriented circuits which are associated to extremal curves of X. As the statement is well-known for the case where X is either a 1-circuit toric variety or a Hirzebruch surface, we can assume without loss of generality, that the extremal curves belong to blow-downs, i.e. the associated oriented circuits are of the form C+ = {i−1, i+1}, C− = {i} for any i ∈ [n]. Now assume that D is in the interior of HC for such an oriented circuit C. Then there exists a bounded chamber PcI in MQ such that ˆ sc (m) to be acyclic, it is necessary that sc (m) ∩ ([n] \ C) = ∅. Let C− = C ∩ sc (m). In order for ∆ {j, k, l} =: D ⊂ [n] represent any other circuit such that D+ = {j, l} for some orientation D of c c c D. The hyperplane arrangement given by the three hyperplanes Hj , Hk , Hl has six unbounded regions, whose signatures contain any subset of {j, k, l} except {j, l} and {k}. In the cases j = i − 2, k = i − 1, l = 1 or j = i, k = i + 1, l = i + 2, PcI must be contained in the region with signature {i}. In every other case PcI must be contained in the region with signature ∅. c In the case, say, {j, k, l} = {i − 2, i − 1, i}, the hyperplane Hi−2 should not cross the bounded c c c chamber related to the subarrangement given by the hyperplanes Hi−1 , Hi , Hi+1 , as else we obtain a chamber whose signature contains {i − 1, i + 1}, but not {i − 2, i}. Then the associated ˆ can never be acyclic. This implies that, if D is in the interior of HC, then subcomplex of ∆ D ∈ HD, where either D = {i − 2, i − 1, i} or D = {i, i +T1, i + 2}. By iterating for every extremal (i.e. every invariant) curve, we conclude that D ∈ i∈[n] HC = nef(X). Analogously, we conclude for D ∈ H−C that D ∈ −nef(X), and the statement follows.  Let {p, q} ⊂ [n] such that lp = −lq . Then these two primitive vectors span a 1-dimensional subvector space of NQ , which naturally separates the set [n] \ {p, q} into two subsets. Definition 3.5.20: Let {p, q} ⊂ [n] such that lp = −lq . Then we denote A1p,q , A2p,q ⊂ [n] the two subsets of [n] \ {p, q} separated by the line spanned by lp , lq . For some fibrational circuit {p, q}, the closure S p,q is a one-dimensional cone in A1 (X)Q which has a unique primitive vector: Definition 3.5.21: Consider{p, q} as P before. Then the closure S p,q is a one-dimensional cone with primitive lattice vector Dp,q := i∈A1p,q li (m)Di , where m ∈ M the unique primitive vector on the ray in MQ with lp (m) = lq (m) = 0 and li (m) < 0 for i ∈ A1p,q .

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Proposition 3.5.22: Let X be a complete toric surface. Then every Ap,q -residual divisor on X is either contained in Anef , or in some Ap,q , or is Anef -residual. c

Proof. For any nef divisor D ∈ −Sp,q , the polytope PD is a line segment such that all Hi intersect this line segment in one of its two end points, depending on whether i ∈ A1p,q or i ∈ A2p,q . This implies that the line spanned by Sp,q is the intersection of all HC , where C ⊂ A1p,q ∪ {p, q} or  C ⊂ A2p,q ∪ {p, q}. Let D be Ap,q -residual and assume that H i X, O(D + rDp,q ) = 0 for all i and for infinitely many r.PWe first show that D ∈ F{p,q} , i.e. that cp +cq = −1 for any torus invariant representative D = i∈[n] ci Di . Assume that cp + cq > −1. Then there exists m ∈ M such that P ′ p, q ∈ / sc (m). By adding sufficiently high multiples of Dp,q such that D + rDp,q = ci Di , we ′ c 1 can even find such an m such that A1 ∪ A2 ⊂ s (m), hence H X, O(D + rDp,q ) 6= 0 for large r and thus D is not Ap,q -residual. If cp + cq < −1, there is an m ∈ M with {p, q} ⊂ sc (m), and  2 by the same argument, we get H X, O(D + rDp,q ) 6= 0 for large r. Hence cp + cq = −1, i.e. D ∈ F{p,q} . This implies that for every m ∈ M either p ∈ sc (m) and q ∈ / sc (m), or q ∈ sc (m) and c p∈ / s (m). Now assume that D ∈ / FC for some C = {i, j, k} ⊂ A1 ∪ {p, q} such that C+ = {i, k} for some orientation. Then there exists some m ∈ M with {i, k} ⊂ sc (m) or {j} ⊂ sc (m). In the ′ first case, as before we can simply add some multiple of Dp,q such that i ∈ sc (m) and i ∈ A2 , ′ / sc (m) or hence sc (m) contains at least two −-intervals. In the second case, we have either p ∈ q∈ / s(m), thus at least two −-intervals, too. Hence D ∈ Ap,q and the assertion follows.  Proposition 3.5.23: Let X be a complete toric surface. Then X has only a finite number of Anef -residual divisors. Proof. We can assume without loss of generality that X is not P2 nor a Hirzebruch surface. Assume there is D ∈ A1 (X) which is not contained in FC for some circuit C = {i − 1, i, i + 1} corresponding to an extremal curve on X. Then there exists a chamber in the corresponding arrangement whose signature contains {i − 1, i + 1}. To have this signature to correspond to ˆ the rest of the signature must contain [n] \ C. Now assume we an acyclic subcomplex of ∆, have some integral vector DC ∈ HC , then we can add a multiple of DC to D such that D is parallel translated to nef(X). In this process necessarily at least one hyperplane passes the critical chamber and thus creates cohomology. Now, D might be outside of FD for some D ∈ C(L) not corresponding to an extremal curve. If the underlying circuit is not fibrational, then D being outside FD implies FC for some extremal circuit C. If D is fibrational and D = {p, q}, then we argue as in Proposition 3.5.22 that D has cohomology. If D is fibrational of cardinality three, the corresponding hypersurface HD is not parallel to any nonzero face of nef(X) and there might be a finite number of divisors lying outside FD but in the intersection of all FC, where C corresponds to an extremal curve.  Proposition 3.5.24: Let X be a complete toric surface. Then X has only a finite number of 0-residual divisors. Proof. Let us consider some vector partition function VP(L, P I) : OI −→ N, for I such that CI does not contain a nonzero subvector space. Let D = i∈[n] ci Di ∈ Ω(L, I) and let PD the polytope in MQ such that m ∈ MQ is in PD iff li (m) < −ci for i ∈ I and li (m) ≥ −ci for i ∈ [n] \ I. For any J ⊂ [n] we denote PD,J the polytope defined by the same inequalities, but only for i ∈ J. Clearly, PD ⊂ PD,J . Let J ⊂ [n] be maximal with respect to the property that PD,J does not contain any lattice points. If J 6= [n], then we can freely move the hyperplanes given by li (m) = −ci for i ∈ [n] \ I such that PD,J remains constant and thus lattice point free.

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T This is equivalent to say that there exists a nonzero D ′ ∈ C∈C(LJ ) HC and for every such D′ the polytope PD+jD′ does not contain any lattice point for any j ∈ Q>0 . Now assume that J = [n]. This implies that the defining inequalities of PD are irredundant and thus there exists a unique maximal chamber in CI which contains D (if I = ∅ this would be the nef cone by 3.5.16) and thus the combinatorial type of PD is fixed. Now, clearly, the number of polygons of shape PD with parallel faces given by integral linear inequalities and which do not contain a lattice point is finite. By applying this to all (and in fact finitely many) cones OI such that CI does not contain a nontrivial subvector space of AQ , we see that there are only finitely many divisors D which are not contained in Anef or Ap,q .  Proof of theorem 3.5.18. By 3.5.19, nef(X) and the Sp,q are indeed the only relevant strata, which by 3.5.22 and 3.5.23 admit only finitely many residual elements. Hence, we are left with the 0-residuals, of which exist only finitely many by 3.5.24.  Example 3.5.25: Figure 3.1 shows the cohomology free divisors on the Hirzebruch surface F3 which is given by four rays, say l1 = (1, 0), l2 = (0, 1), l3 = (−1, 3), l4 = (0, −1) in some coordinates for N . In Pic(F3 ) ∼ = Z2 there are two cones such that H 1 X, O(D) 6= 0 for every  D which is contained in one of these cones. Moreover, there is one cone such that H 2 X, O(D) 6= 0 for every D; its tip is sitting at KF3 . The nef cone is indicated by the dashed lines.

H

H

1

2

H

1

Figure 3.1. Cohomology free divisors on F3 . The picture shows the divisors contained in Anef as black dots. The white dots indicate the divisors in A2,4 . There is one 0-residual divisor indicated by the grey dot. The classification of smooth complete toric surfaces implies that every such surface which is not P2 , has a fibrational circuit of rank one. Thus the theorem implies that on every such surface there exist families of line bundles with vanishing cohomology along the inverse nef cone. For a given toric surface X, these families can be explicitly computed by checking for every C ⊂ A1 ∪ {p, q} and every C ⊂ A2 ∪ {p, q}, respectively, whether the inequalities ( ( ≥ 0 for i ∈ C+ ≥ 0 for i ∈ −C+ ci + li (m) c + l (m) i i < 0 for i ∈ C− , < 0 for i ∈ −C− have solutions  m ∈ M for at least one of the two orientations C, −C of C. This requires to deal with |A13|+2 + |A23|+2 , i.e. of order ∼ n3 , linear inequalities. We can reduce this number to order ∼ n2 as a corollary from our considerations above:

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Corollary 3.5.26: Let C ∈ Ai for i = 1 or i = 2. Then there exist {i, j} ⊂ C such that F{p,q} ∩ FC ⊃ F{p,q} ∩ F{i,j,p} ∩ F{i,j,q} . Proof. Assume first that there exists m ∈ M which for the orientation C of C = {i1 , i2 , i3 } with C+ = {i1 , i3 } which fulfills the inequalities lik (m)+ cik ≥ 0 for k = 1, 3 and li2 (m)+ ci2 < 0. This implies that H 1 X, O(D) 6= 0, independent of the configuration of the other hyperplanes, as long as cp +cq = −1. It is easy to see that we can choose i, j ∈ C such that {i, j, p} and {i, j, q} form circuits. We can choose one of those such that m is contained in the triangle, fulfilling the respective inequalities, and which is not fibrational. For the inverse orientation −C, we can the same way replace one of the elements of C by one of p, q. By adding a suitable positive multiple  of Dp,q , we can rearrange the hyperplanes such that H 1 X, O(D + rDp,q ) 6= 0.  One should read the corollary the way that for any pair i, j in A1 or in A2 , one has only to check whether a given divisor fulfills certain inequalities for triples {i, j, q} and {i, j, p}. It seems that it is not possible to reduce further the number of equations in general. However, there is a criterion which gives a good reduction of cases for practical purposes: P Corollary 3.5.27: Let X be a smooth and complete toric surface and D = i∈[n] ci Di ∈ Ap,q , then for every i ∈ A1 ∪ A2 , we have: ci−1 + ci+1 − ai ci ∈ [−1, ai − 1],

where the ai are the self-intersection numbers of the Di . Proof. The circuit C = {i− 1, i, i+ 1} comes with the integral relation li−1 + li+1 + ai li = 0. So the Frobenius problem for these circuits is trivial and we have only to consider the offset part.  The following example shows that these equalities are necessary, but not sufficient in general: Example 3.5.28: We choose some coordinates on N ∼ = Z2 and consider the complete toric surface defined by 8 rays l1 = (0, −1), l2 = (1, −2), l3 = (1, −1), l4 = (1, 0), l5 = (1, 1), l6 = (1, 2), l7 = (0, 1), l8 = (−1, 0). Then any divisor D = c1 D1 + · · · + c8 D8 with c = (−1, 1, 1, 0, 0, 1, 0, −k) for some k ≫ 0 has nontrivial H 1 , though it fulfills the conditions of corollary 3.5.27.

 An interesting and more restricting case is the additional requirement that also H i X, O(−D) = 0 for all i > 0. One may compare the following with the classification of bundles of type B in [HP06].

Corollary 3.5.29: Let X be a smooth and complete toric surface and D ∈ Ap,q such that  i i H X, O(D) = H X, O(−D) = 0 for all i > 0. Then for every i ∈ A1 ∪ A2 , we have: ( {±1, 0} if ai < −1 ci−1 + ci+1 − ai ci ∈ {−1, 0} if ai = −1, where the ai are the self-intersection numbers of the Di . Proof. For −D, we have cp + cq = 1. Assume that there exists a circuit circuit C with orientation C and C+ = {i, j} and C− = {k}, and morover, some lattice point m such that sc (m) ∩ C = C− . Then we get s−c (−m) ∩ C = C+ . this implies that H 1 X, O(−D) 6= 0. This implies the restriction ci−1 + ci+1 − ai ci ∈ [−1, min{1, ai − 1}].  Note that example 3.5.28 also fulfills these more restrictive conditions.

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3.5.3. Maximal Cohen-Macaulay Modules of Rank One. The classification of maximal Cohen-Macaulay modules can sometimes be related to resolution of singularities, the most famous example for this being the McKay correspondence in the case of certain surface singularities ([GSV83], [AV85], see also [EK85]). In the toric case, in general one cannot expect to arrive at such a nice picture, as there does not exist a canonical way to construct resolutions. However, there is a natural set of preferred partial resolutions, which is parameterized by the secondary fan. Let X be a d-dimensional affine toric variety whose associated convex polyhedral cone σ has dimension d. Denote x ∈ X torus fixed  point. For any Weil divisor D on X, the sheaf OX (D) is MCM if and only if Hxi X, OX (D) for all i < d. It was shown in [BG03] (see also [BG02]) that there exists only a finite number of such modules. ˜ be a toric variety given by some triangulation of σ. The natural map π : X ˜ −→ X Now let X is a partial resolution of the singularities of X which is an isomorphism in codimension two and has at most quotient singularities. In particular, the map of fans is induced by the identity on N and, in turn, induces a bijection on the set of torus invariant Weil divisors. This bijection induces ˜ which can be represented by the identity a natural isomorphism π −1 : Ad−1 (X) −→ Ad−1 (X) n morphism on the invariant divisor group Z . This allows us to identify a torus invariant divisor ˜ Moreover, there are the natural isomorphisms D on X with its strict transform π −1 D on X.  π∗ OX˜ (π −1 D) ∼ = OX (D) and OX˜ (π −1 D) ∼ = π ∗ OX (D) ˇˇ.  Our aim is to compare local cohomology and global cohomology, i.e. Hxi X, OX (D) and  ˜ O ˜ (D) . In general, we have the following easy statement about general (i.e. nonH i X, X regular) triangulations: Theorem 3.5.30: Let X be an affine toric variety of dimension d and D ∈ Ad−1 (X). If D is ˜ −→ X. 0-essential, then Ri π∗ OX˜ (π ∗ D) = 0 for every triangulation π : X Proof. If D is 0-essential, then it is contained in the intersection of all FC , where C ∈ C(L), thus it represents a cohomology-free divisor.  Note that the statement does hold for any triangulation and not only for regular triangulations. We have a refined statement for affine toric varieties whose associated cone σ has simplicial facets: Theorem 3.5.31: Let X be a d-dimensional affine toric variety whose associated cone σ has simplicial facets and let D ∈ Ad−1 (X). If Ri π∗ OX˜ (π ∗ D) = 0 for every regular triangulation ˜ −→ X then OX (D) is MCM. For d = 3 the converse is also true. π:X  Proof. Recall that Hxi X, O(D) m = H i−2 (ˆ σV,m ; k) for some m ∈ M and D ∈ A. We are ˜ of σ such going to show that for every subset I ( [n] there exists a regular triangulation ∆  i ˜ I coincide. This implies that if Hx X, OX (D) that the simplicial complexes σ ˆV,I and ∆ 6= m  i+1 ˜ O ˜ (D) X, 0 for some m ∈ M , then also H 6= 0, i.e. if OX (D) is not MCM, then X m  i ˜ H X, OX˜ (D) 6= 0 for some i > 0. For given I ⊂ [n] we get such a triangulation as follows. Let i ∈ [n] \ I and consider the dual cone σ ˇ . Denote ρi := Q≥0 li and recall that ρˇi is a halfspace which contains σ ˇ and which defines a facet of σ ˇ given by ρ⊥ ∩ σ ˇ . Now we move ρˇi to m + ρˇ, where li (m) > 0. So we obtain a new polytope P := σ ˇ ∩ (m + ρˇ). As ρ⊥ is not parallel to any face of σ ˇ , the hyperplane m + ρ⊥ ˜ of σ which intersects every face of σ ˇ . This way the inner normal fan of P is a triangulation ∆

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has the property that every maximal cone is spanned by ρi and some facet of σ. This implies ˜I = σ ∆ ˆV,I and the first assertion follows.  For d = 3, a sheaf O(D) is MCM iff Hx2 X, O(D) = 0, i.e. H 0 (σV,m ; k) = 0 for every m ∈ M . The latter is only possible if σV,m represents an interval on S 1 . To compare this  ˜ O(D) for some regular triangulation X, ˜ we must show that H 1 (∆ ˜ m ; k) = 0 for with H 2 X, ˜ the corresponding complex ∆m . To see this, we consider some cross-section σ ∩ H, where H ⊂ N ⊗Z R is some hyperplane which intersects σ nontrivially and is not parallel to any of its faces. Then this cross-section can be considered as a planar polygon and σV,m as some ˜ of this connected sequence of faces of this polygon. Now with respect to the triangulation ∆ polygon, we can consider two vertices p, q ∈ σV,m which are connected by a line belonging to the triangulation and going through the interior of the polygon. We assume that p and q have maximal distance in σV,m with this property. Then it is easy to see that the triangulation of σ induces a triangulation of the convex hull of the line segments connecting p and q. Then ˜ m is just the union of this convex hulls with respect all such pairs p, q and the remaining line ∆   ˜ O(D) = 0 for segments and thus has trivial topology. Hence Hx2 X, O(D) = 0 implies H 2 X, ˜ of σ. every triangulation ∆  Example 3.5.32: Consider the 3-dimensional cone spanned over the primitive vectors l1 = (1, 0, 1), l2 = (0, 1, 1), l3 = (−1, 0, 1), l4 = (−1, −1, 1), l5 = (1, −1, 1). The corresponding toric variety X is Gorenstein and its divisor class group is torsion free. For A2 (X) ∼ = Z2 we choose the basis D1 + D2 + D5 , D5 . In this basis, the Gale duals of the li are D1 = (−1, −1), D2 = (2, 0), D3 = (−3, 1), D4 = (2, −1), D5 = (0, 1). Figure 3.2 shows the set of MCM modules in A2 (X) which are indicated by circles which are supposed to sit on the lattice A2 (X) ∼ = Z2 . The picture also indicates the cones CI with vertices −eI , where I ∈ {{1, 3}, {1, 4}, {2, 4}, {2, 5}, {3, 5}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {2, 3, 5}, {2, 4, 5}}. Note that the picture has a reflection symmetry, due to the fact that X is Gorenstein. Altogether, there are 19 MCM modules of rank one, all of which are 0-essential. For C = {l1 , l3 , l4 , l5 }, the group A2 (X)C ∼ = Z ⊕ Z/2Z has torsion. The two white circles indicate modules are contained in the Qhyperplanes D1 + D4 + HC and D2 + D3 + D5 + HC , respectively, but not in the sets D1 + D4 + ZC and D2 + D3 + D5 + ZC , respectively. Some of the OI are not saturated; however, every divisor which is contained in some (−eI + CI ) ∩ Ω(L, I) is also contained in some OI ′ \ Ω(L, I ′ ) for some other I ′ 6= I. So for this example, the Frobenius arrangement gives a full description of MCM modules of rank one.

D3

D5 D2

D1

D4

Figure 3.2. The 19 MCM modules of example 3.5.32.

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Example 3.5.33: To give a counterexample to the reverse direction of theorem 3.5.31 for d > 3, we consider the four-dimensional cone spanned over the primitive vectors l1 = (0, −1, −1, 1), l2 = (−1, 0, 1, 1), l3 = (0, 1, 0, 1), l4 = (−1, 0, 0, 1), l5 = (−1, −1, 0, 1), l6 = (1, 0, 0, 1). The corresponding variety X has 31 MCM modules of rank one which are shown in figure 3.3. Here, with basis D1 and D6 , we have D1 = (1, 0), D2 = (1, 0), D3 = (−1, −2), D4 = (3, 1), D5 = (−2, −2), D6 = (0, 1). There are six cohomology cones corresponding to I ∈  {1, 2}, {3, 5}, {4, 6}, {1, 2, 3, 5}, {1, 2, 4, 6}, {3, 4, 5, 6} .

D1, D2

D4

D6

D3 D5

Figure 3.3. The 31 MCM modules of example 3.5.33. The example features two modules which are not 0-essential, indicated by the grey dots  sitting on the boundary of the cones −eI + CI , where I ∈ {4, 6}, {1, 2, 3, 5} . The white dots denote MCM divisors D, −D such that there exists a triangulation of the cone of X such that ˜ we have H i X, ˜ O(±D) 6= 0 for some i > 0. Namely, we consider on the associated variety X the triangulation which is given by the maximal cones spanned by {l1 , l2 , l4 , l5 }, {l1 , l2 , l4 , l6 }, {l1 , l2 , l5 , l6 }, {l1 , l3 , l4 , l6 }, {l2 , l3 , l4 , l6 }. Figure 3.5.33 indicates the two-dimensional faces of this triangulation via a three-dimensional cross-section of the cone. l2 l3 l4

l6

l5

l1

˜ in example 3.5.33. Figure 3.4. The triangulation for X  We find that we have six cohomology cones corresponding to I ∈ {1, 2}, {3, 5}, {1, 2, 3}, {4, 5, 6}, {1, 2, 3, 5}, {3, 4, 5, 6} . In particular, we have non-vanishing H 1 for the points −D1 − D2 − D3 and for −D4 − D5 − D6 , which correspond to D and −D.

CHAPTER 4

Exceptional sequences of invertible sheaves on rational surfaces 4.1. Introduction The study of derived categories of coherent sheaves on algebraic varieties has gained much attention since the mid-90’s, with some of the main motivations coming from Kontsevich’s homological mirror symmetry conjecture [Kon95] and, evolving from this, the use of derived categories for D-branes in superstring theory [Dou01]. The object one studies is the derived category D b (X) of coherent sheaves over a smooth algebraic variety X defined over some algebraically closed field K. By definition, D b (X) is a categorial framework for the homological algebra of coherent sheaves on X. It turns out that Db (X) carries a very rich structure and encodes information which might not directly be visible from the geometry of X. For an overview we refer to the book [Huy06] and the survey article [Bri06]. However, despite of many interesting and deep results, the theory seems far from being developed enough to make Db (X) an easily accessible object in any sense. A particular open problem is the construction of suitable generating sets, for which the framework of exceptional sequences has been developed by the Seminaire Rudakov [Rud90]: Definition: A coherent sheaf E on X is called exceptional if HomOX (E, E) = K and ExtiOX (E, E) = 0 for every i 6= 0. A sequence E1 , . . . , En of exceptional sheaves is called an exceptional sequence if ExtkOX (Ei , Ej ) = 0 for all k and for all i > j. If an exceptional sequence generates D b (X), then it is called full. A strongly exceptional sequence is an exceptional sequence such that ExtkOX (Ei , Ej ) = 0 for all k > 0 and all i, j. If a full exceptional sequence E1 , . . . , En exists on X and hEi i denotes the minimal triangulated subcategory of D b (X) containing Ei , then hE1 i, . . . hEn i forms a semi-orthogonal decomposition of Db (X), i.e. we have hEj i ⊂ hEi i⊥ for all i > j. Such decompositions naturally arise in birational geometry (see [Orl93], [Kaw08]) and for Fourier-Mukai transforms (see [HvdB07]). Full strongly exceptional sequences provide an even stronger characterization of D b (X) in terms of representation theory of algebras [Hap88]. By theorems of Baer [Bae88] and Bondal [Bon90] for such a sequence there exists an equivalence of categories RHom(T , . ) : Db (X) −→ D b (End(T ) − mod), L where T := ni=1 Ei , which is sometimes called a tilting sheaf. This way the algebra End(T ), at least in the derived sense, represents a non-commutative coordinate system of X. Strongly exceptional sequences have classically been known for the case of Pn (see [Be˘ı78] and [DL85]). However, exceptional or strongly exceptional sequences must not exist in general, and their existence still is an open problem. For instance, on Calabi-Yau varieties it follows from Serre duality that there do not even exist exceptional sheaves. On the other hand, by now, exceptional sequences have been constructed in many interesting cases, including certain types of homogeneous spaces [Kap86], [Kap88], [Kuz05], [Sam07], del Pezzo surfaces and 65

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almost del Pezzo surfaces [Gor89], [KO95], [Kul97], [KN98], and some higher dimensional Fano varieties [Nog94], [Sam05]. In this paper we consider exceptional sequences on smooth complete rational surfaces which consist of invertible sheaves. This special setting is motivated by a conjecture of King [Kin97], which states that on every smooth complete toric variety there exists a strongly exceptional sequence of invertible sheaves. Invertible sheaves on toric varieties can be described in very explicit combinatorial terms and a number of examples were well-known when the conjecture was stated. Also of interest here is the fact that toric varieties can nicely be represented as moduli spaces of certain quiver representations and their universal sheaf is a good candidate for a (partial) tilting sheaf. Examples of strongly exceptional sequences have been given from this point of view in [Kin97] and [AH99] (see also [Bro06], [CS06], [BP08]). Other constructions have been given in [CM04], [CM05], and for toric stacks in [BH08]. Typically, general constructions are only available for very special situations such as iterated projective bundles, or small Picard number. It is known that strongly exceptional sequences of invertible sheaves exist on the toric 3-Fanos, and computer experiments indicate that this is also true for 4-Fanos. However, general existence theorems are only available for exceptional sequences which are not strongly exceptional. So it has been shown in [Hil04] that exceptional sequences of invertible sheaves exist on smooth toric surfaces. The existence of exceptional sequences which do not necessarily consist of invertible sheaves has been shown for general smooth projective toric stacks by Kawamata [Kaw06]. Despite a lot of positive evidence, the existence of strongly exceptional sequences still is an open problem for toric varieties. In [HP06] an example was given of a toric surface which does not admit a strongly exceptional sequence of invertible sheaves, the second Hirzebruch surface iteratively blown up three times. This counterexample at that time seemed somewhat mysterious, in particular because, having Picard number 5, it is surprisingly small. For general rational surfaces there is no bound for the Picard number. This can be shown by well-known examples, such as simultaneous blow-ups of P2 in several points, by which any Picard number can be realized (see Theorem 4.5.9). In the toric case, explicit positive examples with higher Picard numbers were known to the authors, including further blow-ups of the counterexample (see example 4.8.4). So the question is, what is the obstruction for the existence of a (strongly) exceptional sequence of invertible sheaves on a toric or more general rational surface? It turns out that toric surfaces are at the heart of the problem, even for the case of general rational surfaces. The most important structural insight of this paper is the following remarkable observation: Theorem (4.3.5): Let X be a smooth complete rational surface, let OX (E1 ), . . . , OX (En ) be a full exceptional sequence of invertible sheaves on X, and set En+1 := E1 − KX . Then to this sequence there is associated in a canonical way a smooth complete toric  surface with torus invariant prime divisors D1 , . . . , Dn such that Di2 + 2 = χ OX (Ei+1 − Ei ) for all 1 ≤ i ≤ n.

Of course, this theorem deserves a more detailed explanation which will be given below. For the convenience of the reader we want first to present the most important consequences derived from this. Our first main result shows the existence of exceptional sequences in general: Theorem (4.5.6): On every smooth complete rational surface there exists a full exceptional sequence of invertible sheaves.

We point out that for rational surfaces this theorem is not a big surprise and can also be derived from results of Orlov [Orl93]. However, as noted above, an analogous theorem does not hold if we require the sequences to be strongly exceptional. A necessary condition for the existence of a full strongly exceptional sequence seems to be that the surface is not too far away

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from a minimal model. By the Enriques classification, every smooth complete rational surface is a blow-up of the projective plane or some Hirzebruch surface. In fact, we can prove that such sequences exist on a surface which comes from blowing up a Hirzebruch surface once or twice, possibly in several points in every step. Theorem (4.5.9): Any smooth complete rational surface which can be obtained by blowing up a Hirzebruch surface two times (in possibly several points in each step) has a full strongly exceptional sequence of invertible sheaves. In the toric case, we can show that the converse is also true: Theorem (4.8.2): Let P2 6= X be a smooth complete toric surface. Then there exists a full strongly exceptional sequence of invertible sheaves on X if and only if X can be obtained from a Hirzebruch surface in at most two steps by blowing up torus fixed points. Note that the blow-up of P2 at any point is isomorphic to the first Hirzebruch surface. So there is no loss of generality if only blow-ups of Hirzebruch surfaces are considered. In particular, Theorem 4.8.2 implies that the Picard number of a toric surface on which a full strongly exceptional sequence of invertible sheaves exists is at most 14. On the other hand, the example given in [HP06] is a minimal example which does not satisfy the condition of the theorem. Another important aspect of exceptional sequences is their relation to helix theory as developed in [Rud90]. Definition: An infinite sequence of sheaves . . . , Ei , Ei+1 , . . . is called a cyclic (strongly) exceptional sequence if there exists an n such that Ei+n ∼ = Ei ⊗ O(−KX ) for every i ∈ Z and if every winding (i.e. every subinterval Ei+1 , . . . , Ei+n ) forms a (strongly) exceptional sequence. A cyclic exceptional sequence is full if every winding is a full exceptional sequence. Our notion of cyclic strongly exceptional sequences is very close to the geometric helices of [BP94], but we want to point out that these notions do not coincide, as we do not require that our cyclic exceptional sequences are generated by mutations. In fact, if we consider a winding Ei+1 , . . . , Ei+n as the foundation of a helix, then the n-th right mutation of Ei coincides with Ei+n up to a shift in the derived category. By results of [Bon90] a foundation of a helix generates the derived category precisely if any foundation does. Hence a cyclic exceptional sequence is full if and only if it has any winding which is a full exceptional sequence. By a result of Bondal and Polishchuk, the maximal periodicity of a geometric helix on a surface is 3, which implies that P2 is the only rational surface which admits a full geometric helix. Our weaker notion admits a bigger class of surfaces, but still imposes very strong conditions: Theorem (4.5.13): Let X be a smooth complete rational surface on which a full cyclic strongly exceptional sequence of invertible sheaves exists. Then rk Pic(X) ≤ 7. So not even every del Pezzo surface admits such a sequence. However: Theorem (4.5.14): Let X be a del Pezzo surface with rk Pic(X) ≤ 7, then there exists a full cyclic strongly exceptional sequence of invertible sheaves on X. The condition that −KX is ample can be weakened in general. In the toric case we obtain a complete characterization for toric surfaces admitting cyclic strongly exceptional sequences: Theorem (4.8.5 & 4.8.6): Let X be a smooth complete toric surface, then there exists a full cyclic strongly exceptional sequence of invertible sheaves on X if and only if −KX is nef.

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Note that cyclic strongly exceptional sequences have been considered before, most notably in physics literature (see [HHV06], [Asp08], [BP06], [HK06]), but usually under different names. Theorems 4.8.5 and 4.8.6 have been conjectured in this context. The particular interest here comes from the fact that the total space π : ωX → X of the canonical bundle OX (KX ) is a local Calabi-Yau manifold. It follows from results of Bridgeland [Bri05] that a full strongly exceptional sequence E1 , . . . , En on X can be extended to a cyclic strongly exceptional sequence iff the pullbacks π ∗ E1 , . . . , π ∗ En form a sequence on ωX which is almost exceptional in the sense that the π ∗ Ei generate Db (ωX ) and Extk (π ∗ Ei , π ∗ Ej ) = 0 for every i, j and all k > 0 (however, due to the fact that ωX is not complete, we cannot expect that any Hom-groups among the π ∗ Ei vanish). Another interesting observation is that for the toric singularities which arise from L contracting the zero section in ωX , the endomorphism algebras of ni=1 π ∗ Ei give examples for non-commutative resolutions in the sense of van den Bergh [vdB04a], [vdB04b]. Now we give some more technical explanations concerning Theorem 4.3.5 and its consequences. The key idea is astoundingly simple. Let X be a smooth complete rational surface and E1 , . . . , En Cartier divisors on X such that OX (E1 ), . . . , OX (En ) form an exceptional sequence of invertible sheaves. For these sheaves, there are natural isomorphisms ExtkOX OX (Ei ),   OX (Ej ) ∼ = H k X, OX (Ej − Ei ) and therefore it is convenient to bring this exceptional sequence into a normal form by passing to differences. We set Ai := Ei+1 − Ei for 1 ≤ i < n Pn−1 and An := −KX − i=1 Ai , where KX denotes the canonical divisor. The reason for adding An will become clear below. The fact that the Ei form an exceptional sequence then implies  P H k X, OX (− i∈I Ai ) = 0 for every interval I ⊂ [1, . . . , n − 1] and every k > 0. It is an easy consequence of the Riemann-Roch theorem that moreover the Ai have the following properties: (i) Ai .Ai+1 = 1 for 1 ≤ i < n and A1 .An = 1; (ii) P Ai .Aj = 0 for i 6= j, {i, j} = 6 {1, n}, and {i, j} = 6 {k, k + 1} for any 1 ≤ k < n; n (iii) A = −K . X i=1 i

Definition: We call a set of divisors on X which satisfy the conditions (i), (ii), (iii) above a toric system. With respect to a toric system we consider the short exact sequence A

0 −→ Pic(X) −→ Zn −→ Z2 −→ 0, where A maps a divisor class D to the tuple (A1 .D, . . . , An .D). The images l1 , . . . , ln of the standard basis of Zn in Z2 are the Gale duals of A = A1 , . . . , An . It is now an exercise in linear algebra (see Proposition 4.2.7) to show that the li generate the fan of a smooth complete toric surface which we denote Y (A). This means, by passing from E1 , . . . , En via its toric system to the vectors l1 , . . . , ln , we have a canonical way of associating a toric surface to a strongly exceptional sequence of invertible sheaves on any rational surface. This correspondence is even stronger; as Gale duality is indeed a duality, we can as well consider the Ai as Gale duals of the li . But by a standard fact of toric geometry, the Gale duals of the li can be interpreted as the classes of the torus invariant prime divisors D1 , . . . , Dn on Y (A). Hence, we can identify Pic(X) and Pic Y (A) and the respective intersection products in a natural way, such that A2i = Di2 for all i. In particular, note that the set of invariant irreducible divisors forms a toric system for any smooth complete toric surface. Implicitly, toric systems have already shown up in the classical analysis of del Pezzo surfaces. In modern form, this seems first to be written in the first edition of [Man86] (see also [Dem80]). b

bt−1

b

b

2 1 Consider X a t-fold blow-up of P2 , i.e. X = Xt →t Xt−1 → · · · → X1 → P2 . Then we get a

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nice basis H, R1 , . . . , Rt of Pic(X), where H is the pull-back of the class of a line on P2 , and Ri is the pull-back of the exceptional divisor of the blow-up bi . This basis diagonalizes the intersection product of Pic(X), i.e. H 2 = 1, Ri2 = −1 and H.Ri = 0 for all i, and Ri .Rj = 0 for all i 6= j. For simplicity, let us assume that t > 5. Then we construct a graph as follows. For the vertices, we set A0 := H − R1 − R2 − R3 and Ai := Ri − Ri+1 for i = 1, . . . , t − 1 and we draw an edge between Ai and Aj whenever Ai .Aj 6= 0. This way we obtain a graph of type Et which is indefinite for t > 8. For t ≤ 8 it is shown in [Man86] that the set of divisors {D ∈ Pic(X) | χ(−D) = −KX .D = 0} forms a root system which is generated by the Ai . In case of t = 6 this root system represents the symmetries of the famous 27 lines on the cubic surface. The system of divisors A0 , . . . , At−1 is almost a toric system. P We can turn it into a proper toric system by removing A0 and adding At := Rt , At+1 := H − ti=1 Ri , At+2 := H, and At+3 := H − R1 . This toric system always represents an exceptional sequence which is of the form OX , OX (R1 ), . . . , OX (Rt ), OX (H), OX (2H). In case that the bi commute, this sequence is even strongly exceptional. Note that there always are ambiguities concerning the enumeration of the Ai ; we always can try to change it cyclically or even choose the reverse enumeration. This sequence gives an example of an exceptional sequence which is an augmentation of the standard sequence on P2 . On P2 there exists a unique toric system, which is of the form H, H, H. After blowing up once, we can augment this toric system by inserting R1 in any place and subtracting Ri in the two neighbouring positions, i.e., up to symmetries, we obtain a toric system H − R1 , R1 , H − R1 , H on X1 . Continuing with this, we essentially get two possibilities on X2 , namely H − R1 − R2 , R2 , R1 − R2 , H − R1 , H H − R1 , R1 , H − R1 − R2 , R2 , H − R2 . It is easy to see that all of these examples lead to strongly exceptional sequences for almost all enumerations which keep the cyclic order. The only exception being the first one in the case where b2 is a blow-up of an infinitesimal point. Here, we necessarily have to choose the enumeration of the Ai such that An = R1 − R2 . Similarly, on any Hirzebruch surface Fa there exist, in fact infinitely many, toric systems of the form P, sP + Q, P, −(a + s)P + Q with s ≥ −1, which correspond to strongly exceptional sequences. Here, P and Q are the two generators of the nef cone in Pic(Fa ), where P is the class of a fiber of the P1 -fibration Fa → P1 and Q is the generator with Q2 = a. We can extend these toric systems along blow-ups in an analogous fashion. We call toric systems obtained this way standard augmentations (see Definition 4.5.4). It turns out that Theorem 4.8.2 is a consequence of the following characterization of strongly exceptional sequences arising from standard augmentations. Theorem (4.5.11): Let P2 6= X be a smooth complete rational surface which admits a full strongly exceptional sequence whose associated toric system is a standard augmentation. Then X can be obtained by blowing up a Hirzebruch surface two times (in possibly several points in each step). Standard augmentations provide a straightforward procedure which allows to produce strongly exceptional sequences of invertible sheaves on a large class of rational surfaces. It is natural to ask whether it is actually possible to get all such sequences this way. The answer so far is: probably yes. Indeed, Theorem 4.8.2 is a corollary of Theorem 4.5.11 and the following result: Theorem (4.8.1): Let X be a smooth complete toric surface, then every full strongly exceptional sequence of invertible sheaves comes from a toric system which is a standard augmentation.

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Conjecturally, this Theorem should generalize to general rational surfaces. However, our result is based on a rather detailed analysis of cohomology vanishing on toric surfaces which we cannot easily extend to the general case. Moreover, a standard augmentation does not necessarily look like a standard augmentation at the first glance. In the phrase “comes from” in above theorem is hidden a normalization process which must be performed and, as such, is almost obvious (see the end of section 4.5 for details), but whose necessity significantly increases the difficulty of the classification. It turns out that in the toric case all “difficult” strongly exceptional sequences are related to cyclic exceptional sequences. These in turn are easier to understand, but in no case it is a priori clear whether a given strongly exceptional sequence is cyclic. We hope to obtain a more geometric understanding for this in future work. Overview. In section 4.2, after surveying some standard facts on the geometry of smooth complete toric surfaces, we introduce toric systems and explain their relation to toric surfaces. In section 4.3 we derive some elementary properties from cohomology vanishing and show that to every exceptional sequence on a smooth complete rational surface there is associated a toric system. Section 4.4 contains some general results for cohomology vanishing on rational surfaces. Based on this, we prove in section 4.5 our results for exceptional sequences on general rational surfaces, except for Theorem 4.5.11, which is proved in section 4.6. Sections 4.7 to 4.10 are entirely devoted to the case of toric surfaces. In section 4.7 we give a detailed description of cohomology vanishing of divisors on smooth complete toric surfaces. Section 4.8 contains the main results on strongly exceptional sequences on toric surfaces. In sections 4.9 and 4.10 we give a proof of Theorem 4.8.1. Notation and general conventions. For some positive integer l, we denote [l] := {1, . . . , l}. If we use the letter n (or n − 1, n + 1, n + k, etc.), we will usually assume that the elements of [n] are in cyclic order in the sense that we consider [n] as a system of representatives of Z/nZ. In particular, for some i ∈ [n] and some j ∈ Z, we identify i + j with the corresponding class in [n]. If we use some different letter, say t, then we will usually consider the standard total order on the set [t]. Depending on context, we may also consider other partial orders on the set [t]. An interval I ( [n] is a subset I = {i, i + 1, . . . , i + k}, where i ∈ [n], i + k ≤ n and 0 ≤ k < n − 1. A cyclic interval I ( [n] is either an interval or the union I = I1 ∪ I2 of two intervals such that 1 ∈ I1 and n ∈ I2 . For any Z-module K, we will denote KQ := K ⊗Z Q. For some divisor D on a variety X, we will usually omit the subscript X for the corresponding invertible sheafOX (D) if there is no ambiguity for X. We denote  hi (D) := hi O(D) := dim H i X, OX (D) . We will frequently make use of the fact that for any Cartier divisor Don an algebraic surface X and any blow-up b : X ′ → X there are isomorphisms  i ′ ∗ i ∼ H X , b OX (D) = H X, OX (D) for every i ∈ Z.

Acknowledgements. We would like to thank both the Institut Fourier, Grenoble and the Mathematisches Forschungsinstitut Oberwolfach, for their hospitality and generous support. 4.2. The birational geometry of toric surfaces

For general reference on toric varieties, we refer to [Oda88] and [Ful93]. The specifics for toric surfaces are taken from [MO78] and [Oda88]. For Gale transformation, we refer to [GKZ94] and [OP91]. Let X be a smooth complete toric surface defined over some algebraically closed field K. That is, there exists a two-dimensional torus T ∼ = (K∗ )2 acting on X such that T itself is embedded as maximal open and dense orbit in X on which the action restricts to the group multiplication of T . It is clear that every such X is rational. We denote M = Hom(T, K∗ ) ∼ = Z2 and N = Hom(K∗ , T ) ∼ = Z2 the character and cocharacter groups of T , respectively. The toric surface X is completely determined by a collection of

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71

elements l1 , . . . , ln ∈ N with the following properties. We assume that the li are circularly ordered and indexed by elements in [n]. Then for every i ∈ [n] the pair li , li+1 forms a positively oriented basis of N . Moreover, for every such pair there exists no other lk such that lk = αi li + αi+1 li+1 for some nonnegative integers αi , αi+1 . Every pair li , li+1 generates a twodimensional rational polyhedral cone in the vector space NQ , and the collection of faces of all these cones is the fan ∆ associated to X. There is a one-to-one correspondence of 1-dimensional T -orbits in X and the rays in ∆, i.e. the one-dimensional cones, which have the li as primitive vectors. The corresponding orbit closures we denote by Di . Every Di is isomorphic to P1 , and for every i, the divisors Di and Di+1 intersect transversely in the torus fixed point associated to the cone generated by li and li+1 , thus Di .Di+1 = 1. This way, the Di form a cycle of rational curves in X of arithmetic genus 1. Moreover, for every i ∈ [n] there exists the unique relation li−1 + ai li + li+1 = 0, where ai = Di2 ∈ Z is the self-intersection number of Di . Clearly, if just the integers ai are known, we can reconstruct the li from the ai up to an automorphism of N . However, an arbitrary sequence of ai ’s does not necessarily lead to a welldefined smooth toric surface. An admissible sequence a1 , . . . , an is determined by the minimal model program for toric surfaces. Whenever ai = −1 for some i, we can equivariantly blow down the corresponding Di and obtain another smooth toric surface X ′ on which T acts. This surface is specified by a sequence a′1 , . . . , a′i−1 , a′i+1 , . . . a′n (where, up to a cyclic change of enumeration, we can assume that 1 < i < n) such that a′i−1 = ai−1 + 1, a′i+1 = ai+1 + 1, and a′k = ak for k 6= i − 1, i, i + 1. Conversely, an equivariant blow-up at some point Di ∩ Di+1 is described by changing a1 , . . . , ai , ai+1 , . . . an to a1 , . . . , ai−1 , ai − 1, −1, ai+1 − 1, ai+2 , . . . , an . This way, we arrive at the same class of minimal models as in the case of general rational surfaces: Theorem 4.2.1: Every toric surface can be obtained by a finite sequence of equivariant blowups of P2 or some Hirzebruch surface Fa . In particular, the sequences of self-intersection numbers associated to P2 and the Fa are 1, 1, 1 for P2 and 0, a, 0, −a for Fa . Every other admissible sequence a1 , . . . , an can be obtained by successive augmentation of one of these sequences by the aforementioned process. In particular, this implies Proposition 4.2.2: Let XPbe a smooth complete toric surface determined by self-intersection numbers a1 , . . . , an . Then ni=1 ai = 12 − 3n. There is also a local version of above theorem:

Proposition 4.2.3 ([MO78]): Let i < k such that li , lk form a positively oriented basis. Then there exists a sequence of blow-downs from X to a smooth complete toric surface X ′ whose associated primitive vectors are l1 , . . . , li , lk , . . . , ln . The Picard group of X is generated by the T -invariant divisors D1 , . . . , Dn . More precisely, we have a short exact sequence (7)

L

0 −→ M −→ Zn −→ Pic(X) −→ 0,

where L = (l1 , . . . , ln ), i.e. the li are considered as linear forms on M . The i-th element of the standard basis of Zn maps to the rational equivalence class of the divisor Di . There is no canonical choice of coordinates for Pic(X), but there is a very natural and convenient representation for toric divisors if considered as elements in the group of numerical equivalence

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classes of curves N1 (X). Consider the natural pairing on X: N1 (X) ⊗ Pic(X) −→ Z,

(C, D) 7→ C.D,

which is a non-degenerate bilinear form. The pairing is completely specified by the intersection products of the Di among each other, which are given by   ai if i = j, Di .Dj = 1 if j ∈ {i − 1, i + 1},   0 else. D

Denote D := (Di .Dj )i,j=1,...,n the corresponding matrix. Then we have a linear map Zn → Zn whose kernelP is M , the group of numerically trivial T -invariant divisors. Given a T -invariant divisor D := i∈[n] ci Di , its image D(D) is a tuple of the form (d1 , . . . , dn ), where di := di (D) := ci−1 + ai ci + ci+1 = D.Di . If we dualize sequence (7), we get LT

0 −→ Pic(X)∗ −→ Zn −→ N −→ 0,

(8)

where LT denotes the transpose of L. The kernel of LT coincides with the image of D, so that we can identify Pic(X)∗ with N1 (X) in a natural way as subgroups of Zn . So, if considered as a curve, the tuple (d1 , . . . , dn ) is a natural representation of D which does not depend on the choice of a T -invariant representative. Moreover, by sequence (8) we have for any tuple (d1 , . . . , dn ) ∈ N1 (X) that X di li = 0. i∈[n]

By this we can identify N1 (X) with the set of closed polygonal lines in NQ whose segments are given by some multiple of every li . We will make use of this and give some more detail in section 4.7. Note that to determine whether some D is nef, it suffices to test this on the T -invariant divisors. We have: Proposition 4.2.4: Let D a T -invariant divisor on X, then (i) for every i ∈ [n] we have di = deg O(D)|Di ; (ii) D is nef iff di ≥ 0 for every i ∈ [n]. In particular: P Proposition 4.2.5: Denote KX = − ni=1 Di the canonical divisor on X. Then di (KX ) = −ai − 2 for all i. Then −KX is nef iff ai ≥ −2 for all i and −KX is ample iff ai ≥ −1 for i.

Note that on a smooth toric surface an invertible sheaf is ample if and only if it is very ample. There are precisely 16 smooth complete toric surface whose anti-canonical divisor is nef (including the 5 del Pezzo surfaces which admit a toric structure). These are shown in table 1 in terms of the self-intersection numbers ai . In this table, the first four surfaces are given their standard names, the other labels just reflect the length of the sequence a1 , . . . , an . The short exact sequence (7) is an example for a Gale transform. By general properties of Gale transforms, for any subset I of {1, . . . , n}, the set LI := {li | i ∈ I} forms a basis of N iff the complementary set {Di | i ∈ / I} forms a basis of Pic(X), and LI is a minimal linearly dependent set iff the complementary set is a maximal subset of the Di which is contained in a hyperplane in Pic(X). Moreover, we can invert any Gale transform by considering the dual short exact sequence. So by the sequence (8) we get back the li from the Di .

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73

Name self-intersection numbers a1 , . . . , an P2 1, 1, 1 1 0, 0, 0, 0 P × P1 F1 0, 1, 0, -1 0, 2, 0, -2 F2 5a 0, 0, -1, -1, -1 0, -2, -1, -1, 1 5b -1, -1, -1, -1, -1, -1 6a 6b -1, -1, -2, -1, -1, 0 0, 0, -2, -1, -2, -1 6c 6d 0, 1, -2, -1, -2, -2 -1, -1, -2, -1, -2, -1, -1 7a 7b -1, -1, 0, -2, -1, -2, -2 -1, -2, -1, -2, -1, -2, -1, -2 8a 8b -1, -2, -2, -1, -2, -1, -1, -2 -1, -2, -2, -2, -1, -2, 0, -2 8c -1, -2, -2, -1, -2, -2, -1, -2, -2 9 Table 1. The 16 complete smooth toric surfaces whose anti-canonical divisor is nef.

Definition 4.2.6: Let P be a free Z-module of rank n − 2 together with a integral symmetric bilinear form h , i. A sequence of elements A1 , . . . , An in P is called an abstract toric system iff it satisfies the following conditions: (i) hAi , Ai+1 i = 1 for i ∈ [n]; (ii) P hAi , Aj i = 0 for i 6= j and {i, j} = 6 {k, k + 1} for all k ∈ [n]; n (iii) hA , A i = 12 − 3n. i i i=1

Clearly, for any given smooth complete toric surface X, the divisors D1 , . . . , Dn form an abstract toric system in Pic(X) with respect to the intersection form. We show that the data specifying an abstract toric system is equivalent to defining a toric surface. Proposition 4.2.7: Let P , h , i as in definition 4.2.6, A1 , . . . , An an abstract toric system and consider the Gale duals l1 , . . . , ln in N := Zn /P of the Ai . Then N ∼ = Z2 and the l1 , . . . , ln generate the fan of a smooth complete toric surface X with T -invariant irreducible divisors D1 , . . . , Dn such that Di2 = hAi , Ai i for every 1 ≤ i ≤ n. In particular, we can identify P with Pic(X) and h , i with the intersection form on Pic(X). Proof. For n < 3 there is nothing to prove and for n = 3 the statement is easy to see. So we assume without loss of generality that n ≥ 4. We first show that {Aj | j 6= i, i + 1} forms a basis of Pic(X) for every i ∈ [n]. This implies that N ∼ = Z2 and, by Gale duality, that the complementary pairs of li are bases of N . Up to cyclic renumbering, it suffices to show that A1 , . . . , An−2 is a basis of Pic(X). We have hA1 , A2 i = 1, hAn , A1 i = 1 and hAn , A2 i = 0. As h , i is integral, this implies that A1 , A2 generate a subgroup of rank two of P . This subgroup is saturated, i.e. every element in P which can be represented by a rational linear combination of A1 and A2 , can also be represented by an integral linear combination of A1 and A2 . We proceed by induction. Assume that i < n − 2 and that A1 , . . . , Ai are linearly independent and span P a saturated subgroup of rank i of P . For any linear combination B := ij=1 αj Aj , we have hB, Ai+2 i = 0. But hAi+1 , Ai+2 i = 1 and therefore Ai+1 cannot be such a linear combination and thus is linearly independent of A1 , . . . , Ai . From integrality of the bilinear form it follows

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that A1 , . . . , Ai+1 forms a saturated subgroup of P . So by induction A1 , . . . , An−2 is a basis of P. By Gale duality, we thus obtain a sequence of integral vectors l1 , . . . , ln in N ∼ = Z2 , where ⊥ every pair li , li+1 with i ∈ [n] forms a basis of N . Consider the quotient P/Ai ∼ = Z for any i. By ⊥ choosing an appropriate generator of P/Ai , we can identify the images of Ai−1 and Ai+1 with 1 and the image of Ai with ai . If we consider these elements as the Gale duals of li−1 , li , li+1 alone, we see that for every i we have a unique relation li−1 + ai li + li+1 = 0 for ai = hAi , Ai i ∈ Z. It only remains to show that for every lk there do not exist li , li+1 and αi , αi+1 ≥ 0 such that lk = αi li +αi+1 li+1 . As the li , li+1 form bases of N for every i, we see that the ordering (clockwise or counterclockwise) of the li might result in several “windings” until closing up with the final pair ln , l1 . Assume that we partition the li according to such windings, i.e. we group them to W1 = {l1 , . . . , lk1 }, W2 = {lk1 +1 , . . . , lk2 }, . . . , Wr = {lkr−1 +1 , . . . , lkr }, where kr = n. For every two windings Wj , Wj+1 , we get that there exist αj , αj+1 such that l1 = αj lkj + αj+1 lkj+1 . We now add additional rays: first, we add l1j = l1 for every Wj , second we add rays between lkj and l1j and between l1j and lkj+1 such that any two neighbouring rays are lattice bases of N . This way, we obtain a stack of r fans in N , each of which corresponds to a smooth toric surface. We denote n′ the total number of rays after performing this process and a′i the new intersection numbers; then we get by Propositions 4.2.2 and 4.2.3: X X X a′i = ak − 3(n′ − n) = 12r − 3n′ ⇒ ai = 12r − 3n = 12 − 3n, i

i

k

hence r = 1.



So we define: Definition 4.2.8: Let A = A1 , . . . , An be an abstract toric system, then we write Y (A) for the associated toric surface. As we have seen, toric systems provide an alternative way to describe toric surfaces. Assume X is a toric surface, specified by lattice vectors l1 , . . . , ln in N and D1 , . . . , Dn the associated torus invariant divisors, which form a toric system. Then an equivariant blow-down X → X ′ is described by removing some li with li = li−1 + li+1 . This induces an embedding of Pic(X ′ ) in Pic(X) as a hyperplane such that Di .D = 0 for all D ∈ Pic(X ′ ). This corresponds to removing ci , . . . Dn to Pic(X ′ ). More explicitly, for abstract toric systems this Di and projecting D1 , . . . , D can be formulated as:

Lemma 4.2.9: Let A1 , . . . , An be an abstract toric system in P and i such that hAi , Ai i = −1. Then A1 , . . . , Ai−2 , Ai−1 + Ai , Ai+1 + Ai , Ai+2 , . . . An is a toric system as well which is contained in the hyperplane A⊥ i with intersection product h , i|A⊥ . i

Proof. Denote L := (l1 , . . . , ln ) the matrix whose columns are the li , L′ := (l1 , . . . , lbi , . . . , ln ), and consider A := (A1 , . . . , An ) as n-tuple of linear forms on P ∗ . Then the statement is equivalent to describing the map A′ with respect to in the following diagram: 0

// (P ′ )∗ _

A′

0

 // P ∗

A

which is a straightforward computation.

// Zn−1 _  // Zn

L′

L

// Z2

// 0

// Z2

// 0,



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75

So we denote: Definition 4.2.10: Let A1 , . . . , An be an abstract toric system and i such that hAi , Ai i = −1. Then we call A1 , . . . , Ai−2 , Ai−1 + Ai , Ai+1 + Ai , Ai+2 , . . . , An its blow-down. P For a given abstract toric system A, the sum i Ai corresponds to the anti-canonical divisor of Y = Y (A). A small computation shows that the Euler characteristics of the −Ai vanishes: Lemma 4.2.11: Let A = {A1 , . . . , An } be an abstract toric system, then for all i: X 1 Aj , Ai i) = 0. χY (A) (−Ai ) = 1 + (hAi , Ai i − h 2 j

P Proof. We just note that h j Aj , Ai i = hAi−1 , Ai i+hAi , Ai i+hAi+1 , Ai i = 2+hAi , Ai i.  P P Note that in general for two given toric systems A and A′ the sums ni=1 Ai and ni=1 A′i do not coincide. This can most trivially be seen in the case where A′ = −A. Any integral orthogonal transformation maps toric systems to toric systems and in general such transformations do P not leave ni=1 Ai invariant, as we show in the following example. Example 4.2.12: As in the introduction, consider X to be a t-fold blow-up of P2 with H, R1 , . . . , Rt a basis of Pic(X). Denote Ri = {E ∈ Pic(X) | χ(−E) = 0 and − KX .E = i} for every i ∈ Z. It follows from the Riemann-Roch formula that E 2 = i − 2 for every E ∈ Ri (compare Lemma 4.3.3 below). Now, for any i, s ∈ Z with (i − 2)s = −2, and any E ∈ Ri we can define a reflection rE on Pic(X) by setting rE (D) = s(E.D)E + D for any D ∈ Pic(X). Such a reflection clearly respects the intersection product. However, by definition, such a reflection preserves the anti-canonical divisor if and only if E ∈ R0 . If we take the abstract toric system R1 − R2 , R2 − R3 , . . . , Rt−1 − Rt , Rt , H −

t X

Ri , H, H − R1

i=1

from the introduction and apply, say, rR1 to it, where R1 ∈ R1 , then we obtain −R1 − R2 , R2 − R3 , . . . , Rt−1 − Rt , Rt , H + R1 −

t X

Ri , H, H + R1 .

i=2

These divisors add up to rR1 (−KX ) = 3H + R1 −

Pt

i=2 Ri

= −KX + 2R1 .

For constructing and analyzing abstract toric systems, we will need a weaker version: Definition 4.2.13: Let P be a free Z-module of rank n − 2 together with a integral symmetric bilinear form h , i. A sequence of elements A1 , . . . , Ar with r < n in P is called a short toric system if it satisfies the following conditions: (i) hAi , Ai+1 i = 1 for 1 ≤ i < r and hA1 , Ar i = 1; (ii) hAi , Aj i = 0 for i 6= j, {i, j} = 6 {1, r}, and {i, j} = 6 {k, k + 1} for all k ∈ [r − 1]. There are two natural ways for constructing short toric systems from abstract toric systems:

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Example 4.2.14: Let A1 , . . . , An be an abstract toric system, t > 1 and I1 , . . . , It ⊂ [n] a partition of [n] into cyclic intervals such that Ij ∪ Ij+1 (I1 ∪ It , respectively) form a cyclic P interval for every j. Let A′j = k∈Ij Ak , then A′1 , . . . , A′t is a short toric system.

Example 4.2.15: Let X be a smooth complete rational surface and b : X ′ → X a blow-up. If A1 , . . . , An is an abstract toric system in Pic(X) with respect to the intersection form, then b∗ A1 , . . . , b∗ An is a short toric system in Pic(X ′ ). 4.3. Rational surfaces and toric systems Let X be a smooth complete rational surface. From now on we fix n := Pic(X) + 2. Recall that on a rational surface every invertible sheaf is exceptional. For any two   divisors D, E on X, we have natural isomorphisms ExtiOX O(D), O(E) ∼ = H i X, O(E − D) . Let E1 , . . . , En ∈ Pic(X) such that O(E1 ), . . . , O(En ) form an exceptional sequence, then H k X, O(Ei − Ej ) = 0 for all i > j and every k ≥ 0. If, moreover, the sequence is strongly exceptional, we additionally get H k X, O(Ei − Ej ) = 0 for all i, j and all k > 0. This leads to the following definition:

Definition 4.3.1: Let D ∈ Pic(X), then D is called

(i) numerically left-orthogonal to OX if χ(−D) = 0, (ii) left-orthogonal to OX if hi (−D) = 0 for all i, and (iii) strongly left-orthogonal to OX if it is left-orthogonal to OX and hi (D) = 0 for all i > 0. We will usually omit the reference to OX and simply say that D is, e.g. left-orthogonal. The strength of above conditions is completely determined by h1 -vanishing: Lemma 4.3.2: Let D ∈ Pic(X) be numerically left-orthogonal. Then D is left-orthogonal iff h1 (−D) = 0. If −KX is effective, then D is strongly left-orthogonal iff h1 (−D) = h1 (D) = 0. Proof. By assumption χ(−D) = 0. Then clearly h1 (−D) = 0 iff h0 (−D) = h2 (−D) = 0 iff D is left-orthogonal. It remains to show the “strongly” part for h1 (D) = 0. For this we have to show that h2 (D) = 0. By Serre duality, we have h2 (D) = h0 (KX − D). If h0 (KX − D) 6= 0, we get an inclusion h0 (−KX ) ⊂ h0 (−D), but this is impossible, because h0 (−D) = 0 and −KX is effective.  By Riemann-Roch we have χ(D) = 1 + 21 (D 2 − KX .D) for any divisor D, by which we get by symmetrization and anti-symmetrization: χ(D) + χ(−D) = 2 + D 2

and

χ(D) − χ(−D) = −KX .D. By numerical left-orthogonality we have χ(−D) = 1 + 12 (D2 + KX .D) = 0 (compare this also to Lemma 4.2.11), which directly implies: Lemma 4.3.3: Let D, E ∈ Pic(X) numerically left-orthogonal, then (i) χ(D) = −KX .D; (ii) D 2 = χ(D) − 2; in particular, if D is strongly left-orthogonal, then D 2 = h0 (D) − 2; (iii) D+E is numerically left-orthogonal iff E.D = 1 iff χ(D)+χ(E) = χ(D+E); in particular, if D, E, E + D are strongly left-orthogonal, then h0 (D + E) = h0 (D) + h0 (E); (iv) E − D is numerically left-orthogonal iff D.E = χ(D) − 1; in particular, if D, E, E − D are strongly left-orthogonal, then h0 (D) ≤ h0 (E) and D.E = h0 (D) − 1.

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77

Clearly, if O(E1 ), . . . O(En ) is a full exceptional sequence, then n = rk K0 (X) = rk Pic(X)+2 and all the differences Ej − Ei for i > j are left-orthogonal and in particular numerically leftPn−1 orthogonal. We set Ai := Ei+1 − Ei for 1 ≤ i < n and An := −KX − i=1 Ai . Then by Lemma 4.3.3 we get: (1) Ai .Ai+1 = 1 for i ∈ [n]; (2) A 6 {k, k + 1} for some k ∈ [n]; j = 0 for i 6= j and {i, j} = Pi .A n (3) A = −K . X i=1 i Therefore we get an abstract toric system from an P exceptional sequence. Note in general not Pthat n n every abstract toric system can be of this form, as i=1 Ai = −KX implies ( i=1 Ai )2 = 12−3n, but not vice versa, as example 4.2.12 shows. But with this stronger condition, we pass from abstract toric systems to actual toric systems: Definition 4.3.4: Let X a smooth complete rational surface. Then a toric system (on X) is P an abstract toric system A1 , . . . , An ∈ Pic(X) such that ni=1 Ai = −KX . Note that after passing from the E1 , . . . , En to A = A1 , . . . , An , the construction of the toric surface Y (A) is entirely canonical. In particular, we conclude the following remarkable observation: Theorem 4.3.5: Let X be a smooth complete rational surface. Then to any full exceptional sequence of invertible sheaves on X with associated toric system A we can associate in a canonical way a smooth complete toric surface Y (A) with torus invariant prime divisors D1 , . . . , Dn such that Di2 = A2i for every i ∈ [n]. A toric system generates an infinite sequence of invertible sheaves n−1 X

. . . , O(−An ), OX , O(A1 ), O(A1 + A2 ), . . . , O(

Ai ), O(−KX ), O(−KX + A1 ), . . .

i=1

If some subsequence of length n of this sequence is a strongly exceptional sequence, we will follow the convention that the toric system that this sequence can be written as Pn−1is enumerated such P OX , O(A1 ), O(A1 + A2 ), . . . , O( i=1 Ai ). In particular, i∈I Ai is strongly left-orthogonal for every interval I ⊂ [n − 1]. In general we will assume nothing about the strong left-orthogonality P of An . If the toric system gives rise to a cyclic strongly exceptional sequence, then i∈I Ai is strongly left-orthogonal for every cyclic interval I ⊂ [n]. Definition 4.3.6: We say that a toric system A1 , . . . , An is (cyclic, strongly) exceptional if Pn−1 the associated sequence of invertible sheaves OX , O(A1 ), . . . , O( i=1 Ai ) generates a (cyclic, strongly) exceptional sequence. Note that a priori a toric system and the conditions on cohomology vanishing do not completely determine the ordering of the Ai . In particular, if A1 , . . . , An is a cyclic (strongly) exceptional toric system, then so is An , . . . , A1 . If A1 , . . . , An is a (strongly) exceptional toric system, then so is An−1 , . . . , A1 , An . 4.4. Left-orthogonal divisors on rational surfaces Any smooth complete rational surface X can be obtained by a sequence of blow-ups X = b

bt−1

bt−2

b

1 t Xt −→ X0 , where X0 is either P2 or some Hirzebruch surface Fa . If Xt−1 −→ Xt−2 −→ · · · −→ we fix the sequence of morphisms bt , . . . , b1 , we obtain a natural basis of Pic(X) with respect to this sequence as follows. If X0 = P2 , we denote as before H the hyperplane class of P2 , and for

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every bi , we denote Ri the class of the associated exceptional divisor in Pic(Xi ). For simplicity, we identify H and the Ri with their pullbacks in Pic(X). Every blow-up increases the rank of the Picard group by one and the pullback yields an inclusion of Pic(Xi−1 ) into Pic(Xi ) as a hyperplane. Then Ri is additional generator, which is orthogonal to Pic(Xi−1 ) with respect to the intersection product. We have the following relations: H 2 = 1,

Ri2 = −1,

H.Ri = 0 for all i,

and Ri .Rj = 0

for all i 6= j.

In particular, we have t = rk Pic(X) − 1. So, in the case where X is a blow-up of P2 , we easily get a basis of Pic(X) which diagonalizes the intersection product. In the case where X0 = Fa for some a ≥ 0, we start with a basis P, Q of Pic(Fa ) as before, and by the same process, we obtain a basis P, Q, R1 , . . . , Rt of Pic(X), where t = rk Pic(X) − 2. Here, the most convenient choice for our purpose is P, Q to be the integral generators of the nef cone in Pic(Fa )Q such that P 2 = 0 and Q2 = a. So we get P 2 = 0, Q2 = a, P.Q = 1, Ri2 = −1, P.Ri = Q.Ri = 0 for all i, and Ri .Rj = 0 for all i 6= j. Often our arguments below do not depend on the choice of X0 , and for simplicity we will often leave this choice implicit and assume that t = n − 3 or t = n − 4 as it fits. Definition 4.4.1: Let D ∈ Pic(X), then we denote the projection of D to Pic(Xi ) by (D)i . The projection (D)i just is ‘forgetting’ the coordinates Rt , Rt−1 , . . . , Ri+1 , i.e. if D = P P P αP +βQ+ tj=1 γj Rj or D = βH + tj=1 γj Rj , respectively, then (D)i = αP +βQ+ ij=1 γj Rj P or (D)i = βH + ij=1 γj Rj , respectively. By Lemma 4.3.2, left-orthogonality is determined by numerical left-orthogonality and h1 vanishing. Our strategy to understand (strongly) left-orthogonal divisors will be to start with h1 -vanishing and then to establish numerical left-orthogonality. For this, we first need a couple of lemmas related to h0 - and h1 -vanishing. Lemma 4.4.2: Let E be an irreducible (−1)-divisor and X ′ the surface obtained from blowing down E. If D is the pullback to X of some divisor on X ′ , then for every k ∈ Z, we have deg O(D + kE)|E = −k. Proof. For k ∈ Z consider the short exact sequence 0 −→ O(D + (k − 1)E) −→ O(D + kE) −→ OE (D + kE) −→ 0.  Then, for the Euler characteristics, we get χ OE (D+kE) = χ(D+kE)−χ(D+(k−1)E) = 1−k, where the latter equality follows from Riemann-Roch and D.E = 0. Hence OE (D + kE) ∼ = OE (−k) and the assertion follows.  We use this to investigate h0 - and h1 -vanishing. If a divisor has nonzero h1 , then so has its preimage under blow-up. For h0 and h2 , we have the opposite picture: Lemma 4.4.3: Let D and E as in Lemma 4.4.2. (i) If h0 (D) = 0, then h0 (D + kE) = 0 for all k ∈ Z. (ii) If h1 (D) 6= 0, then h1 (D + kE) 6= 0 for all k ∈ Z. (iii) If h2 (D) = 0, then h2 (D + kE) = 0 for all k ∈ Z. Proof. For k = 0 there is nothing to prove. If k > 0, we do induction on k. Consider the short exact sequence 0 −→ O(D + (k − 1)E) −→ O(D + kE) −→ OE (D + kE) −→ 0.

4.4. LEFT-ORTHOGONAL DIVISORS ON RATIONAL SURFACES

79

 By lemma 4.4.2, we have deg O(D + kE)|E = −k and therefore h0 O(D + kE) = 0. So by the long exact cohomology sequence we get h0 (D + (k − 1)E) = h0 (D + kE), h1 (D + (k − 1)E) ≤ h1 (D + kE), and h2 (D + (k − 1)E) ≥ h2 (D + kE). For (i), we have by induction assumption h0 (D + (k − 1)E) = 0 and so h0 (D + kE) = 0. For (ii), we have by induction assumption h1 (D + (k − 1)E) > 0 and so h1 (D + kE) > 0. For (iii), we have by induction assumption h2 (D + (k − 1)E) = 0 and so h2 (D + kE) = 0. For k < 0, we do induction from k + 1 to k. In this case, we consider the short exact sequence 0 −→ O(D + kE) −→ O(D + (k + 1)E) −→ OE (D + (k + 1)E) −→ 0.  So deg O(D + (k + 1)E)|E = −k − 1 ≥ 0 and therefore h1 O(D + (k + 1)E) = 0. Then by the long exact cohomology sequence, we get h0 (D + kE) ≤ h0 (D + (k + 1)E), h1 (D + kE) ≥ h1 (D +(k +1)E), and h2 (D +kE) = h2 (D +(k +1)E). For (i), we have by induction assumption h0 (D + (k + 1)E) = 0 and so h0 (D + kE) = 0. For (ii), we have by induction assumption h1 (D + (k + 1)E) > 0 and so h1 (D + kE) > 0. For (iii), we have by induction assumption h2 (D + (k + 1)E) = 0 and so h2 (D + kE) = 0.  Definition 4.4.4: Let D ∈ Pic(X) (D)0 6= 0. Then we call D ∈ Pic(X) pre-left-orthogonal  with 1 0 with respect to X0 iff h (−D)0 = h (−D) = 0, and strongly pre-left-orthogonal if it is preleft-orthogonal and h1 (D) = 0.

Note the little twist that for pre-left-orthogonality we do not just require h0 -vanishing, but instead have conditions on X0 . This makes the following an immediate consequence of Lemma 4.4.3: Corollary 4.4.5: If D is pre-left-orthogonal, then so is (D)i for i = 1, . . . , t. If D is a pre-left-orthogonal divisor on Xt−1 , then in general D + γt Rt will only be pre-leftorthogonal for a few possible values of γt . The following lemma gives some sufficient conditions.

Lemma 4.4.6: Let D ∈ Pic(X) and k ≥ l ≥ 0. If D − kRt is pre-left-orthogonal, then D − lRt is also pre-left-orthogonal. If D − kRt is strongly pre-left-orthogonal, then so is D − lRt . Proof. We do both cases by induction on l, starting with l = k. For l = k, there is nothing to show. Also (D − kRt )0 = (D − lRt )0 , so there is nothing to show for h0 . Assume now that k > l > 0 and D − lRt is pre-left-orthogonal. We consider the short exact sequence 0 −→ O(−D + (l − 1)Rt ) −→ O(−D + lRt ) −→ ORt (−D + lRt ) −→ 0.

 By lemma 4.4.2 we have deg OE (−D + lRt ) = −l < 0, and thus h0 ORt (−D + lRt ) = 0. Then by the long exact cohomology sequence h1 (−D + (l − 1)Rt ) ≤ h1 (−D + lRt ) = 0 and the first assertion follows by induction. If D − lE is strongly pre-left-orthogonal, we consider the following short exact sequence 0 −→ O(D − lRt ) −→ O(D − (l − 1)Rt ) −→ ORt (D − (l − 1)Rt ) −→ 0. Again, by lemma 4.4.2 he have deg ORt (D − (l − 1)Rt ) = l − 1 ≥ 0 and therefore h1 ORt (D − (l − 1)Rt ) = 0. Then by the long exact cohomology sequence, we have 0 = h1 (D − lRt ) ≥ h1 (D − (l − 1)Rt ) ≥ 0 and the second assertion follows by induction.  Now we classify (strongly) pre-left-orthogonal divisors on P2 and on the Fa . Denote H the class of a line on P2 . As the condition of h1 -vanishing is vacuous for invertible sheaves on P2 , we trivially observe:

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Proposition 4.4.7: A divisor on P2 is (pre-)left-orthogonal iff it is strongly (pre-)left-orthogonal. The pre-left-orthogonal divisors are given by kH, where k > 0, and the left-orthogonal divisors are H, 2H. For the case of a Hirzebruch surface Fa , we choose P, Q as before and the following statements can be seen rather straightforwardly, for instance by using toric methods as in [HP06], [Per07]. Proposition 4.4.8: The pre-left-orthogonal divisors on a Hirzebruch surface are: (i) on F0 : P + kQ, kP + Q for k ∈ Z, kP + lQ for k, l ≥ 2; (ii) on Fa , with a > 0: P , kP + Q for k ∈ Z, kP + lQ for k ≥ 1 − a and l ≥ 2; A pre-left-orthogonal divisors is strongly pre-left-orthogonal iff it is not of the type P + kQ or kP + Q for k < −1 or of type kP + lQ for l ≥ 2 and k < max{−1, 1 − a}. Proposition 4.4.9: Let Fa be a Hirzebruch surface. (i) If a = 0, then the left-orthogonal divisors are given by P + kQ, kP + Q for k ∈ Z. (ii) If a > 0, then the left-orthogonal divisors are given by P , kP + Q for k ∈ Z, and (1 − a)P + 2Q. (iii) Left-orthogonal divisors of type kP + Q or P + kQ are strongly left-orthogonal iff k ≥ −1. Divisors of type (1 − a)P + 2Q are strongly left-orthogonal iff a ≤ 2. In coordinates chosen with respect to a minimal model X0 , the anti-canonical divisor on X can be written as −KX = 3H −

t X

Ri

or

i=1

−KX = (2 − a)P + 2Q −

t X

Ri , respectively.

i=1

For X0 = P2 and some divisor D = βH + formulas for the Euler characteristics of D: 

Pt

i=1 γi Ri ,

we get by Riemann-Roch the following

 X  β+2 γi χ(D) = − 2 2 i   X  β−1 γi + 1 χ(−D) = − , 2 2

(9) (10)

i

where we write we get: (11) (12)

x 2



= 12 x(x − 1) for any x ∈ Z. For X0 = Fa and D = αP + βQ +   X  β+1 γi χ(D) = (α + 1)(β + 1) + a − 2 2 i   X  β γi + 1 χ(−D) = (α − 1)(β − 1) + a − 2 2 i

Pt

i=1 γi Ri ,

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If χ(−D) = 0, we obtain linear equations for χ(D) = −KX D in either coordinates: X χ(D) = 3β + γi i

χ(D) = 2α + (2 + a)β +

X

γi .

i

We now look at the case where (D)0 = 0. In this case, we have to take into account the relative configuration of Ri and Rj . Definition 4.4.10: Assume i, j > 0 and denote xj and xi the points on Xj−1 and Xi−1 , respectively, which are blown up by the maps bj and bi . We define a partial order ≥ on the set {R1 , . . . , Rt } by setting Ri ≥ Ri for every i and Rj ≥ Ri if j > i and bi ◦ · · · ◦ bj−1 (xj ) = xi . Now we get: Proposition 4.4.11: Let D ∈ Pic(X) such thatP(D)0 = 0. Then D is left-orthogonal if there exists i ∈ [t] and S ⊂ [t] \ {i} such that D = Ri − j∈S Rj and Ri  Rj for all j ∈ S. Moreover, D is strongly left-orthogonal iff it is of the form Ri for some i ∈ [t] or of the form Ri − Rj such that Ri and Rj are incomparable with respect to the partial order ≥. Proof. Note that P for (D)0 = 0, by Lemma 4.4.3 (iii), we can always assume that h2 (D) = = 0. Let D = i γi Ri , then χ(−D) = 0 by formula (10) or (12) yields: X γj + 1 = 1. 2

h2 (−D)

j

But then there is precisely one i ∈ [t] with γi ∈ {1, −2} and γj ∈ {0, −1} for all other j. If γi = −2, we consider Ri as irreducible divisor on Xi and we consider the following part of a long exact cohomology sequence: X X X    H 1 Xi , OXi ( Rj ) −→ H 1 Xi , OXi (2Ri + Rj ) −→ H 1 Xi , O2Ri (2Ri + Rj ) −→ 0 j∈S

j∈S

j∈S

 P P P for some S ⊂ [i]. As χ( j∈S Rj ) = 1 = h0 ( j∈S Rj ), we get h1 OXi ( j∈S Rj ) = 0 and thus   P P h1 OXi (2Ri + j∈S Rj ) = h1 O2Ri (2Ri + j∈S Rj ) . By lemma 4.4.3 we can assume without P loss of generality that i ≥ j for all j ∈ S. Then we get O2Ri (2Ri + j∈S Rj ) ∼ = O2Ri (2Ri ) and    P 1 we compute χ O2Ri (2Ri ) = χ O(2Ri ) − 1 = −1 and thus h OX0 (2Ri + j∈S Rj ) 6= 0. P So we are left with divisors of the form Ri − j∈S Rj for some S ⊂ [t]. By Serre duality, we  P P P have h2 (−Ri + j∈S Rj ) = h0 (KX +Ri − j∈S Rj ) ≤ h0 (KX +Ri − j∈S Rj )0 = h0 (KX )0 ) = P 0. If there exists k ∈ S such that Ri ≥ Rk , then Rk − Ri is effective, and −Ri + j∈S Rj is P a sum of effective divisors and therefore h0 (−Ri + j∈S Rj ) 6= 0. If there exists k ∈ S such that Ri and Rk are incomparable, then we may assume that this k is minimal with P respect to ≥. Then h0 (Rk − Ri ) = 0, and by lemma 4.4.3 we can conclude that h0 (−Ri + j∈S Rj ) = 0. The remaining possibility is that Rj ≥ Ri for all j ∈ S. In that case, denote Ei the strict transform on X of the exceptional divisor of the blow-up bi . P Then there exists Ti ⊂ [t] such that P Ei is rationally equivalent to Ri − j∈Ti Rj . Then −Ri + j∈S Rj is rationally equivalent to P P P 0 j∈S\Ti Rj − j∈Ti \S Rj −Ei . If any of S\Ti or Ti \S are empty, we have h (−Ri + j∈S Rj ) = 0. Otherwise,P if any Rk , Rl wit k ∈ S \ Ti and l ∈ Ti \ S are incomparable, then h0 (Rk − Rl ) = h0 (−Ri + j∈S Rj ) = 0. If not, we choose k ∈ Ti \ S. Then there exists Tk ⊂ [t] such that

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P Ek = Rk − l∈Tk Rl and we iterate our previous argument until we get the difference of two P incomparable Rj or we can write −Ri + j∈S Rj as the inverse of an effective divisor. j ∈PS with Ri ≥ Rj , we can now conclude together with χ(−Ri + P So, unless there exists i (−R + R ) = 0 that h all i. This shows the first assertion. For strong j i j∈S j∈S Rj ) = 0 for P left-orthogonality, we necessarily need χ(Ri − j∈S Rj ) ≥ 0, which is the case iff S is empty or S = {j} for some j 6= i. A divisor Ri always is strongly left-orthogonal. For Ri − Rj we have χ(Ri − Rj ) = 0, and h1 (Ri − Rj ) = 0 is equivalent to h0 (Ri − Rj ) = 0. But this is in turn is equivalent to incomparability of Ri and Rj .  For (D)0 6= 0, we have the following statement: P Proposition 4.4.12: If D = (D)0 + i γi Ri is left-orthogonal and (D)0 is pre-left-orthogonal, then γi ≤ 0 for all i. γk +1 Proof. Assume χ(−D) = 0 and γ > 0 for some k, then χ(−D + γ R ) = > 0. k k k 2  0 0 As h (−D)0 = 0, we also have h (−D + γk Rk ) = 0. Therefore we have χ(−D + γk Rk ) = h2 (−D + γk Rk ) − h1 (−D + γk Rk ) > 0, hence h2 (−D + γk Rk ) > 0. But by Serre duality, h2 (−D) = h0 (KX + D) ≥ h0 (KX + D − γk Rk ) = h2 (−D + γk Rk ) > 0, which is a contradiction to the left-orthogonality of D, and the assertion follows.  Remark 4.4.13: Note that in the case where D is strongly left-orthogonal but (D)0 is not strongly pre-left-orthogonal, this implies that h0 (D)0 = 0 and therefore h0 (D) = 0. But then −D is left-orthogonal, too, and (−D)0 is strongly pre-left-orthogonal. We now consider some special cases concerning proposition 4.4.12.

Lemma 4.4.14: Let X be a smooth complete rational surface, D a very ample and strongly ˜ → X in four points x1 , x2 , x3 , x4 , where left-orthogonal divisor on X. Consider a blow-up b : X x1 and x2 are on X and x3 and x4 are infinitesimal points lying over x1 and x2 , respectively. ˜ then the divisors Denote R1 , . . . , R4 the pullbacks of the exceptional divisors of b to Pic(X), ˜ D − Ri and D − Ri − Rj with i 6= j are strongly left-orthogonal on X. Proof. It follows directly from our previous discussions that the divisors D − Ri and D − Ri − Rj are left-orthogonal. It remains to show that h1 (D − Ri ) = h1 (D − Ri − Rj ) = 0. By lemma 4.3.3 (iii) we know χ(D − Ri ) = χ(D) − 1 and χ(D − Ri − Rj ) = χ(D) − 2. So it suffices to show that h0 (D − Ri − Rj ) < h0 (D − Ri ) < h0 (D) for any i 6= j. But this is an immediate consequence of [Har77], V.4, Remark 4.0.2 and preceding remarks.  4.5. Exceptional sequences of invertible sheaves on rational surfaces We first show that cyclicity for exceptional sequences of invertible sheaves is no additional condition: Proposition 4.5.1: Let X be a smooth complete rational surface. Then every exceptional sequence of invertible sheaves is cyclic. Proof. Let A1 , . . . , An be an exceptional toric system. Then for every interval I ⊂ [n − 1] we have hi (−AI ) = 0 for every i. By Serre duality, we get hi (−AI ) = hi (KX + A[n]\I ) = h2−i (−A[n]\I ) = 0 for every i. So AJ is left-orthogonal for every cyclic interval J of [n] and A1 , . . . , An corresponds to a cyclic exceptional sequence. 

4.5. EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES

83

On P2 , there is a unique toric system which gives rise to a cyclic strongly exceptional sequence, but, as we will see for the case of Hirzebruch surfaces, Proposition 4.5.1 does not hold for strongly exceptional sequences in general. Recall that P, Q are generators of the nef cone of the Hirzebruch surface Fa , where P 2 = 0, Q2 = a, and P.Q = 1. Proposition 4.5.2: On a Hirzebruch surface Fa there are the following toric systems: (i) P, sP + Q, P, −(a + s)P + Q for s ∈ Z for any a; (ii) − a2 P + Q, P + s(− a2 P + Q), − a2 P + Q, P − s(− a2 P + Q) for s ∈ Z and a even. Toric systems of type (i) are always exceptional. They are strongly exceptional for s ≥ −1, where A4 = −(a + s)P + Q. They are cyclic strongly exceptional iff s ≥ −1 and a + s ≤ 1. Toric systems of type (ii) are almost never exceptional. The exceptions are for a = 0, where type (ii) is symmetric to type (i) by exchanging P and Q, and for a = 2 and s = 0, which then coincides with a toric system of type (i) and is cyclic strongly exceptional. Proof. Any toric system must represent a Hirzebruch surface. Therefore, for any toric system A1 , A2 , A3 , A4 we can assume that A21 = A23 = 0 and A22 = −A24 = −b for some b ∈ Z. So for a general element αP + βQ with α, β ∈ Z, the equations (αP + βQ)2 = 0 and χ(−αP − βQ) = 0 have always the solution α = 1, β = 0. If a is even, we get a second solution, α = − a2 and β = 1. The condition A1 .A3 = 0 can only be fulfilled if A1 = A3 = P , or if A1 = A3 = − a2 P + Q. In the first case, using A1 .A2 = A1 .A4 = 1 and A2 .A4 = 0, we get that A2 = sP + Q and A4 = −(a + s)P + Q for some s ∈ Z which indeed form a toric system way for every s ∈ Z. In the second case with a even, we similarly compute that A2 = P + s(− a2 P + Q) and A4 = P − s(− a2 P + Q) for some s ∈ Z. The classification of exceptional sequences (cyclic or strong) among these follows by inspection of the classification of (strongly) left-orthogonal divisors of proposition 4.4.9.  Remark 4.5.3: From Proposition 4.5.2 follows that for a toric system A on a Hirzebruch surface Fa , the associated Hirzebruch surface Y (A) is isomorphic to Fb , where b − a is even. As in the previous section, we assume that a sequence of blowups X = Xt −→ · · · −→ X0 is fixed, where X0 is P2 or some Fa , together with a corresponding basis of Pic(X), either H, R1 , . . . , Rt if X0 ∼ = P2 , or P, Q, R1 , . . . , Rt if X0 ∼ = Fa . Any toric system A = A1 , . . . , An−t+i on some Xi pulls back to a short toric system on X in the sense of Definition 4.2.13 (see Example 4.2.15). Such a short toric system can easily be extended to a toric system by using the Ri+1 , . . . , Rt as follows. For any i + 1 ≤ j1 ≤ t we denote A1 the sequence A1 , . . . , As−1 , As − Rj1 , Rj1 , As+1 − Rj1 , As+2 , . . . , An−t+i , which augments A at some position s. Note that this augmentation is understood in the cyclic sense, i.e. we do not exclude s = n − t + i. If i = t − 1, then this sequence is a toric system on X; otherwise, it is again a short toric system. Inductively, for 1 < k < t − i we can in the same way augment Ak−1 to a short toric system Ak by some Rjk for jk ∈ {i + 1, . . . , t} \ {jl | 1 ≤ l < k} and finally we arrive at a toric system At−i . Of course, At−i also depends on the positions at which the Ak have been augmented. A toric system obtained this way in general cannot be interpreted as successive augmentation via pullbacks from the Xj with i < j < t as we have not imposed any condition on the ordering of the jk . We will see below that the interesting augmentations which are obtained this way are precisely those which are augmentations via pullbacks.

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Definition 4.5.4: We call an exceptional toric system on P2 or Fa a standard toric system. On a smooth complete rational surface X, we call a toric system which is the augmentation of a standard toric system a standard augmentation. A standard augmentation is admissible if it P contains no element of the form Ri − j∈S Rj such that Rj ≤ Ri for some j ∈ S.

Note that the condition of admissibility isP precisely the condition of Proposition 4.4.11 on left-orthogonality of divisors of the form Ri − j∈S Rj . This condition implies that a standard augmentation is admissible iff there exists a bijection j : [t] → [t], k 7→ jk such that Rk ≥ Rl iff Rjk ≥ Rjl . Then we can rearrange the ordering of the blow-ups accordingly such that X = Xjt → · · · → Xj1 → X0 and the augmentation then can be considered as an successive augmentation along these blow-ups. The following proposition shows that this way we get many exceptional sequences in the form of standard augmentations. Proposition 4.5.5: Every standard augmentation yields a full exceptional sequence on X iff it is admissible. Proof. Let A = A1 , . . . , An be the augmented sequence. If X0 = P2 , we can renumber P this sequence such that An is of the form H − i∈S Ri for some subset S of [t]. We claim P that A = A1 , . . . , An−1 yield an exceptional sequence iff it is admissible. That is, every AI := i∈I Ai for some non-cyclic interval I ⊂ [n − 1] is left-orthogonal iff A is admissible. Clearly, every such P AI is numerically left-orthogonal. We have two cases. First, lH − i∈T Ri with T ⊂ [t] and P P l ∈ {1, 2}. P By Serre duality we get h2 (−lH + i∈T Ri ) = h0 (−(3 − l)H + i∈T / Ri ) = 0 and thus lH − Pi∈T Ri is left-orthogonal (without any condition on admissibility). Second, we have AI = Ri − i∈T Ri with T ⊂ [t], which is left-orthogonal by proposition 4.4.11 iff Rj  Ri for all j ∈ T . In particular, all AI are of this form iff A is admissible. P If X0 = Fa , we can renumber the sequence such that An is of the form P Q−(a+n)P − i∈S Ri for some subset S of [t]. Then for AI we have three cases. First, P − i∈T Ri with T ⊂ [T ]. By P P P R ) = 0 and so P − − Serre duality we get h2 (−P + i∈T Ri ) = h0 (−2Q−(1−a)P i i∈T Ri i ∈T / P is left-orthogonal. Second, we have Q + nP − i∈T Ri with T ⊂ [T ] and n ∈ Z. Again, by Serre P P duality, wePget h2 (−Q − nP + i∈T Ri ) = h0 (−Q − (2 − n − a)P / Ri ) = 0 and thus P − i∈T Q + nP − i∈T Ri is left-orthogonal. Third, we have AI = Ri − i∈T Ri with T ⊂ [t], which is left-orthogonal by proposition 4.4.11 iff Rj  Ri for all j ∈ T . In particular, all AI are of this form iff A is admissible. We have seen now that a standard augmentation is admissible iff all AI are left-orthogonal. It follows directly from the results of [Orl93] that standard augmentations are full.  So, by observing that we can lift any standard sequence on some X0 to an admissible standard augmentation on X, the following is an immediate consequence of Proposition 4.5.5: Theorem 4.5.6: Every smooth complete rational surface has a full exceptional sequence of invertible sheaves. Let us denote bi : Xi −→ Xi−1 the i-th blow-up in the sequence X = Xt → · · · → X0 . We assume that bi can be partitioned into two sets S1 := {b1 , . . . , bs } and S2 := {bs+1 , . . . bt } for 1 < s ≤ t such that the bi within Sl for l ∈ {1, 2} commute. In other words, we assume that X can be obtained from P2 or Fa by two times simultaneously blowing up (possibly) several points.

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4.5. EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES

Theorem 4.5.7: With above assumptions on X and X0 = P2 , the following is a full strongly exceptional toric system: Rs , Rs−1 − Rs , . . . , R1 − R2 , H − R1 , H − Rs+1 , Rs+1 − Rs+2 , . . . , Rt−1 − Rt , Rt , H −

t X

Ri .

i=1

P

Proof. We have to check that i∈I Ai is strongly left-orthogonal for every interval I ⊂ P [n − 1]. Here we have A1 = Rs and An = H − ti=1 Ri . There are precisely four types of divisors which can be represented in this way, namely Ri , Ri − Rj for Ri , Rj incomparable, H, 2H, H − Ri and 2H − Ri − Rj for i 6= j. The divisors H, 2H are clearly strongly left-orthogonal. The left-orthogonality of Ri and Ri − Rj follows from proposition 4.4.11, the left-orthogonality of H − Ri and 2H − Ri − Rj from Lemma 4.4.14. The toric system clearly is an admissible standard augmentation and so from Proposition 4.5.5 it follows that the resulting exceptional sequence is full.  Analogously, we get: Theorem 4.5.8: With above assumptions on X and X0 = Fa for some a ≥ 0 and n ≥ −1, the following is a full strongly exceptional toric system: Rs , Rs−1 − Rs , . . . , R1 − R2 , P − R1 , nP + Q, P − Rs+1 , Rs+1 − Rs+2 , . . . , Rt−1 − Rt , Rt , −(a + n)P + Q −

t X

Ri .

i=1

P

Proof. Here, i∈I Ai is of the form Ri , Ri − Rj for Ri , Rj incomparable, P , nP + Q with n ≥ −1, P − Ri , nP + Q − Ri for n ≥ 0, and nP + Q − Ri − Rj for n ≥ 1. The divisors P , nP + Q clearly are strongly left-orthogonal (see Lemma 4.4.9). The left-orthogonality of Ri and Ri − Rj follows from Proposition 4.4.11, the left-orthogonality of nP + Q − Ri and nP + Q − Ri − Rj from Lemma 4.4.14. The cases P − Ri and Q − Ri are clear because P and Q are globally generated. Also, the toric system is an admissible augmentation of a standard sequence and so from proposition 4.5.5 it follows that the resulting exceptional sequence is full.  The following theorem is an immediate consequence of Theorem 4.5.8. Theorem 4.5.9: Any smooth complete rational surface which can be obtained by blowing up a Hirzebruch surface two times (in possibly several points in each step) has a full strongly exceptional sequence of invertible sheaves. Remark 4.5.10: Note that for the existence of strongly exceptional sequences it suffices to consider X0 = Fa for some a ≥ 0, as every blow-up of P2 factorizes through a blow-up of F1 . Nevertheless, as we will see later on, for cyclic strongly exceptional sequences it will be advantageous also to consider augmentations coming from P2 . The converse of Theorem 4.5.9 is true for strongly exceptional sequences coming from standard augmentations: Theorem 4.5.11: Let P2 6= X be a smooth complete rational surface which admits a full strongly exceptional standard augmentation then X can be obtained by blowing up a Hirzebruch surface two times (in possibly several points in each step). We prove this theorem in section 4.6.

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Remark 4.5.12: We will see in Theorem 4.8.1 that in the toric case every full strongly exceptional sequence of invertible sheaves is equivalent to a strongly exceptional standard augmentation which implies (Theorem 4.8.2) that a toric surface different from P2 admits such a sequence iff it can be obtained by blowing up a Hirzebruch surface at most two times. So, in a sense, the existence of a full strongly exceptional sequence of invertible sheaves can be considered as a geometric characterization of a surface. Presumably, Theorem 4.8.1 should generalize to all rational surfaces, but at present it is not clear to us whether the procedure of sections 4.7 to 4.10 can be generalized in an effective way. The following theorem gives a strong constraint on the existence of cyclic strongly exceptional sequences of invertible sheaves on rational surfaces in general: Theorem 4.5.13: Let X be a smooth complete rational surface on which a full cyclic strongly exceptional sequence of invertible sheaves exists. Then rk Pic(X) ≤ 7. Proof. Let A = A1 , . . . , An be the associated toric system. As every Ai is strongly leftorthogonal, it follows that χ(Ai ) ≥ 0 for every i. Therefore by Proposition 4.2.5 the anticanonical bundle of the associated toric surface Y (A) must be nef. From  the classification of such toric surfaces (see table 1) it follows that rk Pic(X) = rk Pic Y (A) ≤ 7.  In particular, Theorem 4.5.13 implies that not even every del Pezzo surface has a cyclic strongly exceptional sequence of invertible sheaves. However, if rk Pic(X) ≤ 7, we have the following positive result:

Theorem 4.5.14: Let X be a del Pezzo surface with rk Pic(X) ≤ 7, then there exists a full cyclic strongly exceptional sequence of invertible sheaves on X. Proof. Recall that a del Pezzo surface is either P1 × P1 or a blow-up of P2 in at most 8 points (see [Dem80]). The case P1 × P1 is clear from Proposition 4.5.2. For the other cases, by our assumptions it suffices to assume that X is a blow-up of P2 in at most 6 points x1 , . . . , x6 . Moreover, it suffices to only consider the maximal case, i.e. rk Pic(X) = 7 and the cases of smaller rank will follow immediately. We first give an example for a cyclic exceptional toric system and then show that it is cyclic strongly exceptional. We fix a blow-down X → P2 and denote R1 , . . . , R6 the exceptional divisors and H the class of a line on P2 . Then by Proposition 4.5.5 the following is a full cyclic exceptional sequence: H − R1 − R2 − R5 , R2 , R1 − R2 , H − R1 − R3 − R4 , R4 , R3 − R4 , H − R3 − R5 − R6 , R6 , R5 − R6 . To show that a toric system A1 , . . . , A6 is cyclic P strongly exceptional, we have to show that for every cyclic interval I ⊂ [6] the sum AI := i∈I Ai is strongly left-orthogonal. There are several possible cases what AI can be. First, if AI = Ri for some i ∈ [6] or AI = Ri − Rj for some i 6= j ∈ [6], strong left-orthogonality follows from Proposition 4.4.11. The next cases are of the form H − Ri , H − Ri − Rj and H − Ri − Rj − Rk , respectively, where i, j, k pairwise distinct. Analogous to the arguments in the proof of 4.4.14, we have to discuss the existence of base points. As H is very ample, its associated complete linear system does not have base points. So h0 (H − Ri ) < h0 (H), and we conclude as in the proof of 4.4.14 that H − Ri is strongly left-orthogonal. For any two xi , xj , we can find a line on P2 which does pass through xi but not through xj . So, the linear system |H − Ri | is base point free and H − Ri − Rj is strongly left-orthogonal for any i 6= j. The divisor H − Ri − Rj is not base point free. Its base points lie on the line connecting xi and xj . But as X is del Pezzo, none of the other xk lie on this line. So we have h0 (H − Ri − Rj − Rk ) < h0 (H − Ri − Rj ) and thus H − Ri − Rj − Rk is

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87

P strongly left-orthogonal. Similarly, using [Har77], V.4, Corollary 4.2, we see that P2H − i∈S Ri is strongly left-orthogonal for any S ( [6]. The next cases are of the form 3H − i∈S Ri , where S ⊆ [6] and |S| ≥ 4. As |S| < 7, it follows from [Har77], V.4, Proposition 4.3,P that these are strongly left-orthogonal, too. The remaining cases are of the form 3H − 2Ri − k6=i,j Rk with P i 6= j ∈ [6]. By [Har77], V.4, Proposition 4.3, 3H − k6=j Rk has no base points, therefore P P P h0 (3H − 2Ri − k6=i,j Rk ) < h0 (3H − k6=j Rk ) and 3H − 2Ri − k6=i,j Rk is strongly leftorthogonal  Remark 4.5.15: Note that for a del Pezzo surface X with rk Pic(X) ≤ 7 the toric system of the type as given in the proof of Theorem 4.5.14 in general is not the only possibility. It is an exercise to write down all possible admissible standard augmentations for X0 = P2 and to check the conditions whether the resulting toric system is cyclic and strong. For example, for X del Pezzo, the strongly exceptional toric systems as given in Theorem 4.5.7 are cyclic iff t ≤ 3. Moreover, it follows from the proof of Theorem 4.5.14 that the conditions on X can be weakened in general. Though the toric system given in the proof does require that no three points are collinear, it admits a configuration of 6 points lying on a conic and certain configurations of infinitely near points. We will see in Theorems 4.8.5 and 4.8.6 that at least in the toric case the existence of such sequences is equivalent to −KX nef. We conclude this section with some more technical properties of strongly exceptional sequences. As before, we assume that a sequence of blow-downs to a minimal surface X0 is chosen. First we consider parts of a toric system which are “vertical” with respect to X0 : Lemma 4.5.16: LetPA1 , . . . , Ak ∈ Pic(X) such that Ai .Ai+1 = 1 for 1 ≤ i < k and Ai .Aj = 0 else such that AI := i∈I Ai is strongly left-orthogonal and (AI )0 = 0 for every interval I ⊂ [k]. Then this system is, up to reversing the order of the Ai , of one of the following shapes: (i) A1 = Ri1 − Ri2 , A2 = Ri2 − Ri3 , . . . , Ak = Rik − Rik+1 , (ii) A1 = Ri1 − Ri2 , A2 = Ri2 − Ri3 , . . . , Ak−1 = Rik−1 − Rik , Ak = Rik ,

where the Ril are pairwise incomparable. Proof. By proposition 4.4.11 every AI must be of the form Ri or Ri − Rj for some i, j ∈ [t] such that Ri and Rj are incomparable. Moreover, (Rip − Riq ).(Ris − Rit ) = 1 iff either q = s and p 6= t or q 6= s and p = t. Moreover, (Rip − Riq ).(Ris − Rit ) = 0 iff {p, q} ∩ {s, t} = ∅. This readily implies that the sequence A1 , . . . , Ak must be of one of the above forms.  For the parts of a toric system which are not vertical to Pic(X0 ), we would like to have a normal form. Let O(E1 ), . . . , O(En ) be a strongly exceptional sequence and A1 , . . . , An its associ- P ated toric system. One of the requirements is that Hom O(Ei ), O(Ej ) = H 0 X, O(− i−1 k=j Ai ) = 0 for i > j. So, clearly, for any 1 ≤ i < n with χ(Ai ) = 0, we can exchange Ei and Ei+1 such that O(E1 ), . . . , O(Ei+1 ), O(Ei ), . . . , O(En ) also forms a strongly exceptional sequence. The toric system then becomes: A1 , . . . , Ai−2 , Ai−1 + Ai , −Ai , Ai+1 + Ai , Ai+2 , . . . , An . We introduce the following notion with this operation in mind. Definition 4.5.17: Let A = A1 , . . . , An be a toric system. If A gives rise to a (cyclic) strongly exceptional sequence, we say that A is in normal form with respect to X0 if (Ai )0 is either zero or strongly pre-left-orthogonal for every 1 ≤ i < n (for every 1 ≤ i ≤ n, respectively).

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Assume that A gives rise to a strongly exceptional sequence and is not in normal form, i.e. there exists some Ai with 1 ≤ i < n such that (Ai )0 is non-trivial and not strongly pre-leftorthogonal. This implies that χ(Ai ) = h0 (Ai ) = 0 and there are no homomorphisms between O(Di−1 ) and O(Di ). In fact, there exists a maximal interval I ⊂ [n − 1] containing i such that Hom O(Dk ), O(Dl ) = 0 for every k, l ∈ I. Clearly, any reordering of the Dk with k ∈ I is a strongly exceptional sequence, too. We are going to show that every strongly exceptional sequence comes, up to such reordering, from a toric system in normal form. Proposition 4.5.18: Let X be a smooth complete rational surface and X0 a minimal model for X. Then any (cyclic) strongly exceptional sequence of invertible sheaves on X can be reordered such that the associated toric system is in normal form with respect to X0 . Proof. Let O(E1 ), . . . , O(En ) be a strongly exceptional sequence and A = A1 , . . . , An its associated toric system. As remarked above, for any interval [k, . . . , l + 1] ⊂ [n] such that χ(Ai ) = 0 for every k ≤ i ≤ l, we can exchange the positions of any two O(Ei ), O(Ej ) with i, j ∈ I. In particular, if we want to move O(El+1 ) to the leftmost position, it is easy to see that the toric system becomes . . . , Ak−1 ,

l X i=k

Ai , −

l X

Ai , Ak+1 , . . . , Al−1 , Al + Al+1 , Al+2 , . . .

i=k+1

Let 1 ≤ l < n be minimal such that (Al )0 is non-trivial and not strongly pre-left-orthogonal. Then exchanging O(El+1 ) with O(El ) changes the toric system to . . . , Ai−2 , Ai−1 +Ai , −Ai , Ai+1 + Ai , Ai+2 , . . . such that (−Al )0 is strongly pre-left-orthogonal. Now possibly (Ai−1 + Ai )0 is no longer strongly pre-left-orthogonal. In this case we iterate moving O(El+1 ) to the left. This process eventually stops, because of one of two reasons. First, O(El+1 ) ends up at the most left P P position and we are getting − li=1 Ai , A1 , . . . , Al−1 , Al +Al+1 , Al+2 , . . . , An + li=1 Ai . Second, P O(El+1 ) is at (k + 1)-th position, but ( li=k Ai )0 is strongly pre-left-orthogonal. Consequently, after moving O(El+1 ), the smallest 1 ≤ l′ < n such that (Al′ )0 is non-trivial and not strongly pre-left-orthogonal is strictly bigger than l. So, by iterating this exchange process, we end up with a toric system in normal form. If O(E1 ), . . . , O(En ) is a cyclic strongly exceptional sequence, we are free to cyclically change the enumeration of the Ai . In particular, from the general classification of toric surfaces, it follows that there cannot be a cyclic interval I ⊂ [n] of length bigger than n − 3 such that h0 (Ai ) = 0 for every i ∈ I. Moreover, if A is not in normal form, we can choose the enumeration of the Ai the way that, if (Al )0 is non-trivial and not strongly pre-left-orthogonal, then h0 (A1 ) > 0 and we choose l < k < n minimal such that h0 (Ak ) > 0. This implies that P P ( pi=1 Ai )0 and ( ki=q Ai )0 are strongly pre-left-orthogonal for every 1 ≤ p < k and every 1 < q ≤ k. So the part A1 , . . . , Ak is in normal form. We iterate this and eventually all of A will be in normal form.  4.6. Proof of Theorem 4.5.11 Assume first that X0 = P2 and A0 = H, H, H. If we blow up X1 → X0 , then there is, up to cyclic change of enumeration, only one possible augmentation A1 = H − R1 , R1 , H − R1 , H. But X1 is isomorphic to F1 and if we choose the usual generators P , Q of the nef cone of X1 as a basis of Pic(X1 ), we get P = H − R1 , Q = H. In these coordinates we have A1 = P, Q− P, P, Q, which is the unique cyclic strongly exceptional standard toric system on F1 . So, to prove the theorem it suffices to consider standard toric systems coming from Hirzebruch surfaces according

4.6. PROOF OF THEOREM 4.5.11

89

to the classification of Proposition 4.5.2. We assume that X is obtained by a sequence of blowups X = Xt → · · · → X0 of a Hirzebruch surface X0 ∼ = Fa and denote P, Q, R1 , . . . , Rt the corresponding basis of Pic(X). For any divisor D we denote bs(D) the base locus of the complete linear system |D|. Note that for any effective divisor D the condition χ(−D) = 0 is equivalent to the arithmetic genus of D being zero. It is straightforward to check that in this case the underlying reduced divisor Dred also has arithmetic genus zero. So, because h0 (Ri ) = 1 for every i, the divisor class Ri is represented by a unique, possibly non-reduced, effective divisor of arithmetic genus zero and bs(Ri ) coincides with the support of this divisor whose arithmetic genus is also zero. The image of bs(Ri ) in X0 is contained in some fiber of the ruling Fa → P1 which we denote by fi and which represents the divisor class P . For any Ri we denote Ei the strict transform on X of the corresponding exceptional divisor on Xi . By abuse of notation we also use Ei for the strict transforms on the Xj with j ≥ i. Any effective divisor D whose support contains Ei can be written D = D ′ + ni Ei where D ′ is effective and does not have any component with support Ei . We call ni the multiplicity of Ei with respect to D. By abuse of notion we will also sometimes call ni the multiplicity of Ri . We recall that the Ri form a partially ordered set. The maximal elements have the property that Ei2 = −1. Any maximal chain of Ri contains precisely one maximal element. All maximal elements are incomparable and can be blown down simultaneously. In the nicest cases we will see that the maximal length of maximal chains will be at most two and that X can be blown down to X0 in two steps. However, the most part of our analysis in this section will be concerned with the cases where there exist maximal chains of bigger length. In general there will be only very few of these and, if such chains exist, we will have to look for some other way to blow down to some minimal model X0′ which might not coincide with X0 . For this we will need exceptional divisors which do not coincide with one of the Ei . These exceptional divisors can be the strict transform of some fiber fi or, in the case X0 ∼ = F1 , of the unique divisor on X0 with self-intersection −1. Note in the sequel we will consider the case where blow-ups are only over a fixed fiber f . This will be without loss of generality, because in our conclusion at the end of this section we will make use of the fact that fi 6= fj implies that Ri and Rj are incomparable. Lemma 4.6.1: We use notation as before. (i) For any i, the divisor class P − Ri is strongly left-orthogonal and bs(P − Ri ) contains bs(Rj ) for every Rj with fj = fi and Ri  Rj . (ii) If the multiplicity of Ri with respect to the total transform of fi is greater than 1, then bs(P − Ri ) contains bs(Rj ) for every Rj with fj = fi . (iii) For any i 6= j, the divisor class P − Ri − Rj is strongly left-orthogonal iff either fi 6= fj or Ri and Rj are comparable (say, Ri ≤ Rj ) and bs(P − Ri ) does not contain bs(Rj ). Proof. Clearly we have χ(−P + Ri ) = 0, hk (−P + Ri ) = 0 for all k, and χ(P − Ri ) = 1. To show that P − Ri is strongly left-orthogonal we need only to show that h0 (P − Ri ) = 1. But this follows from the fact that the divisor class P is nef and therefore base-point free and hence h0 (P − Ri ) = h0 (P ) − 1 = 1. The divisor class P − Ri is nontrivial and its base locus is a curve of arithmetic genus zero which projects to fi . The total transform of fi is a representative of P in Pic(X) and contains the base loci of all the Rj with fj = fi . By subtracting Ri from P , we at most (but not necessarily) cancel the base loci of those Rj with Ri ≤ Rj and (i) follows. If the multiplicity of Ri with respect to the total transform of fi is greater than 1, then the multiplicities of all Ej with Ri ≤ Rj with respect to P is strictly smaller than their multiplicities with respect to Ri . Therefore bs(P − Ri ) also contains bs(Rj ) for Ri ≤ Rj and (ii) follows.

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From (i) it follows that statement (iii) essentially is a case distinction for determining when bs(Rj ) is not contained in the base locus of P − Ri .  Lemma 4.6.2: Consider the divisor Q on X0 ∼ = F1 and some strongly left-orthogonal divisor class Q − Ri − Rj on X with Ri , Rj incomparable and f := fi = fj . Then bs(Q − Ri − Rj ) contains the total transform of f . Proof. The class Q is the pullback of the class of lines in P2 . Denote p the image of Ri in X0 , then we can identify the linear system |Q − Ri | with the set of lines passing through the image of p in P2 . If Rj lies over some other point of f than p, then bs(Q − Ri − Rj ) fixes two points on f and thus contains f . If Rj also lies over P , then we first observe that bs(Q − Ri ) contains bs(Rk ) for all k 6= i and Rk ≤ Ri . So, the condition that Q − Ri − Rj is strongly left-orthogonal implies that Ri is minimal and hence Ri ≤ Rj , which is a contradiction.  Lemma 4.6.3: Let A1 , . . . , An a strongly exceptional toric system on X which is a standard augmentation of P, sP + Q, P, −(a + s)P + Q with s ≥ −1 for some choice of X0 ∼ = Fa such that fi = fj for all i, j. Assume that Ak , Ak+1 , . . . Al for some 1 ≤ k < l < n is a subsequence of the toric system which contains the two slots around one of the P , i.e. (Ak−1 )0 , (Al+1 )0 ∈ / {0, P }, (Ap )0 = P for one k ≤ p ≤ l, and (Aq )0 = 0 for all k ≤ q ≤ l with q 6= p. Then Ak , . . . , Al is, up to possible order inversions, of one of these forms: (i) Ril−k , Ril−k−1 − Rik−l , . . . , Ri2 − Ri3 , Ri1 − Ri2 , P − Ri1 , where the Rij are pairwise incomparable; (ii) Ri1 , P − Ri1 − Ri2 , Ri2 − Ri3 , . . . , Ril−k−1 − Rik−l , Rik−l , where Ri1 ≤ Rij and the Rij are pairwise incomparable for j > 1. Proof. After the first augmentation we get Ri1 , P − Ri1 . In the second step, we can extend this sequence in the middle, or to the left, or to the right. By extending in the middle, we get Ri1 − Ri2 , Ri2 , P − Ri1 − Ri2 which implies that bs(Ri2 ) ∈ / bs(Ri1 ) ∪ bs(P − Ri1 ), where the right hand side coincides with the total transform of a fiber fi on X, which is not possible. By extending to the left, we get Ri2 , Ri1 − Ri2 , P − Ri1 with the necessary condition that bs(Ri1 ) ∩ bs(Ri2 ) = ∅ and therefore Ri1 , . . . , Ri2 are incomparable. By iterating to the left, we obtain that the Rij are pairwise incomparable and therefore we arrive at the form (i). If we extend to the right instead, we get Ri1 , P − Ri1 − Ri2 , Ri2 and by Lemma 4.6.1 (iii) Ri1 , Ri2 must be comparable. In the next step, we extend without loss of generality to the right and get Ri1 , P − Ri1 − Ri2 , Ri2 − Ri3 , Ri3 where Ri1 , Ri3 are comparable and Ri2 , Ri3 are incomparable. If we extend to the left in the next step, this implies that the pairs Ri1 , Ri4 and Ri2 , Ri3 are incomparable, but Ri2 and Ri3 are comparable to Ri1 and Ri4 respectively, which is not possible. So, we can continue extending only to the right and we inductively obtain that the Rij are pairwise incomparable for j > 1 and Ri1 is comparable with every Rij . If l − k > 2, this implies that Ri1 ≤ Rij for every j > 1.  Now we consider standard augmentations starting from a standard sequence P, sP +Q, P, −(s+ a)P + Q with s ≥ −1 on X0 . For this, we have four “slots”, in which we can insert the Ri successively. The augmented toric system is of the form A1 , . . . , An , where possibly An is only left-orthogonal but not strongly left-orthogonal. For (An )0 , there are four possibilities, namely (An )0 = 0, (An )0 = P , (An )0 = sP + Q and (An )0 = −(s + a)P + Q. We will first consider the last case. Proposition 4.6.4: Let A = A1 , . . . , An be a strongly exceptional toric system which is a standard augmentation of the toric system P, sP + Q, P, −(s + a)P + Q with s ≥ −1 on X0 ∼ = Fa

4.6. PROOF OF THEOREM 4.5.11

91

such that (An )0 = −(s + a)P + Q and fi = fj for all i, j. Then X can be obtained from blowing up a Hirzebruch surface two times (in possibly several points in each step). Proof. We denote f the distinguished fiber such that f = fi for all i ∈ [t]. Because (An )0 = −(s + a)P + Q the toric system has two subsequences which are of the form as stated in Lemma 4.6.3. This implies that there is a partition of the set {R1 , . . . , Rt } into two subsets S1 := {Ri1 , . . . , Rir }, S2 := {Rj1 , . . . , Rjs } such that, if nonempty, the elements in each of these subsets either are (i) incomparable or (ii) Ri1 ≤ Rik and the Rik incomparable for all k > 1 (Rj1 ≤ Rjk and the Rjk incomparable for all k > 1, respectively). If both S1 and S2 are empty, there is nothing to prove. If one of S1 , S2 is empty, then the length of a maximal chain of comparable elements among the Ri is at most two and the proposition follows. So we assume that S1 and S2 both are nonempty. If S1 and S2 both satisfy property (i), then again the length of a maximal chain of comparable elements among the Ri is at most two and the proposition follows. If both satisfy property (ii), then we have two cases. The first is that Ri1 , Rj1 both are minimal. Then again the length of a maximal chain of comparable elements among the Ri is at most two. The second case is that only one of these, say Ri1 , is minimal and Ri1 = R1 without loss of generality. On X1 we have f 2 = −1 and we can choose to either blow down R1 or f . If we choose f , then we obtain another of basis for Pic(X1 ) given by P ′ , Q′ , R1′ , where P ′ = P , R1′ = P − R1 and Q′ = Q + δP − R1 , where δ = 1 if R1 corresponds to a blow-up of a point on the zero section of the fibration Fa → P1 , and δ = 0 otherwise. If we complete this basis to a basis of Pic(X) by using the Ri with i > 1, the sequence Ri1 , P − Ri1 − Ri2 , Ri2 − Ri3 , . . . , Rir−1 − Rir , Rir becomes P ′ − R1′ , R1′ − Ri2 , Ri2 − Ri3 , . . . , Rir−1 − Rir , Rir with R1′ , Ri2 , . . . Rir pairwise incomparable. So we have reduced to the case that S1 satisfies property (i) and S2 satisfies property (ii). Moreover, we can assume that Rj1 is not minimal as otherwise we can choose another basis as we did before and reduce to the case that both S1 and S2 satisfy property (i). In the remaining case, the length of a maximal chain of comparable Ri is either two or three. If it is two, the proposition follows. In the case where it is three, we assume without loss of generality that A is an augmentation of a strongly exceptional toric system on X3 with R1 ≤ R2 ≤ R3 such that R1 ∈ S1 and R2 , R3 ∈ S2 . Then the divisor P − R2 − R3 is strongly left-orthogonal and by Lemma 4.6.1 (ii) it follows that the multiplicity of R2 with respect to the total transform of f is one. In particular, R2 does not come from a blow-up of the intersection of f with E1 on X1 . If we now go back to X, then the Rik are incomparable with R1 and hence with R2 , because R1 ≤ R2 . Thus the Rik are minimal. So, by blowing down simultaneously all Ei with Ei2 = −1 (which includes E3 ) on X, we arrive at the surface X2 . Here, we have f 2 = −1 and E22 = −1. So, we can blow-down these two divisors simultaneously and arrive at  some Hirzebruch surface X0′ . If s + a > 1, it follows by Lemma 4.4.3 that necessarily (An )0 = −(s + a)P + Q. If s + a ≤ 1, then possibly (An )0 ∈ {0, P } and the standard toric system P, sP + Q, P, −(s + a)P + Q must be cyclic strongly exceptional on Fa for which, by Proposition 4.5.2, there are only four possibilities. Our first step will be to reduce these to one. Proposition 4.6.5: Let A = A1 , . . . , An a toric system on X with (An )0 6= −(s + a)P + Q. Then there exists a sequence of blow-downs X = Xt′ → · · · X0′ such that X0′ ∼ = F1 and A is an augmentation of the toric system P ′ , Q′ , P ′ , Q′ − P ′ on X0′ . Proof. As argued before, A necessarily is an augmentation of a cyclic strongly exceptional standard sequence. In particular, X0 ∼ = Fa with 0 ≤ a ≤ 2. If a = 1, there is nothing to prove. If a = 0, there are, up to symmetry by exchanging P and Q, two such toric systems, P, Q, P, Q and P, P + Q, P, −P + Q. If we consider the blow-up X1 → X0 , then in the first case, there exists,

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up to cyclic reordering and order inversion, only one possible augmentation which is given by P −R1 , R1 , Q−R1 , P, Q which is a cyclic strongly exceptional toric system on X1 . If we consider some projection X0 → P1 such that P represents a general fiber, then the divisor P − R1 is rationally equivalent to the strict transform under the blow-up and has self-intersection (−1). If we blow down this divisor, we obtain X1 → X0′ ∼ = F1 . If we denote P ′ , Q′ the corresponding divisors in Pic(F1 ), then we get a change of coordinates in Pic(X1 ) via P = P ′ , Q = Q′ − R1′ , and R1 = P ′ − R1′ . In this basis the toric system is given as R1′ , P ′ − R1′ , Q′ − P ′ , P ′ , Q′ − R1′ and the assertion follows in this case. We proceed similarly in the second case. As h0 (P − Q) = 0 and (An )0 6= P − Q by assumption, the only possible augmentation (up to cyclic reordering and order inversion) on X1 is given by P − R1 , R1 , P + Q − R1 , P, −P + Q. By the same change of coordinates as before we get R1′ , P ′ − R1′ , Q′ , P ′ , Q′ − P ′ − R1′ and the assertion follows for this case. Now assume that a = 2. Then the only cyclic strongly exceptional toric system is given by P, Q − P, P, Q − P and the only possible augmentation on X1 is given by P − R1 , R1 , Q − P − R1 , P, Q − P . The base locus of the complete linear system of the divisor Q − P consists of one fixed component which is the zero section of the fibration F2 → P1 . Therefore, if X1 is a blow-up on the zero section, we have h0 (Q − P − R1 ) = h0 (Q − P ) = 2 and Q − P − R1 is not strongly left-orthogonal and thus necessarily (An )0 = Q − P which is a contradiction to our assumptions. So we can assume without loss of generality that X1 is a blow-up of X0 at some point which is not on the zero section. In this case we can conclude as before that there exists a blow-down to X0′ ∼ = F1 and a corresponding change of coordinates P = P ′ − R1′ , Q = Q′ + P ′ − R1′ , and R1 = P ′ − R1′ such that the toric system is represented as R1′ , P ′ − R1′ , Q′ − P ′ , P ′ , Q′ − R1′ which is of the required form.  Proposition 4.6.6: Let A = A1 , . . . , An be a strongly exceptional toric system which is a standard augmentation of the toric system P, Q, P, −P + Q on X0 ∼ = F1 . Then X can be obtained by blowing up a Hirzebruch surface two times (in possibly several points in each step). Proof. We will only consider the case (An )0 ∈ {0, P }. Otherwise, the result follows from Proposition 4.6.4. We will denote b the zero section (respectively its strict transform) of the P1 -fibration X0 → P1 with b2 = −1 on X0 . Note that in some steps below we will have to blow-down b to arrive at some convenient minimal model X0′ . Strictly speaking, this would require us not only to consider blow-ups of a fixed fiber f but rather the general case. However, in these few cases this would only increase the number of case distinctions without changing the arguments. So we will keep the assumption that all blow-ups lie above one distinguished fiber f . First note that h0 (−P + Q) = 1 and therefore any divisor of the form −P + Q − Ri − Rj cannot be strongly left-orthogonal. This together with the condition (An )0 6= −P + Q implies that we can use at most one of the two slots around −P + Q in the toric system P, Q, P, −P + Q for augmentations. Moreover, for any (An )0 , we can assume that the augmentations in the two slots surrounding one of the P ’s are strongly left-orthogonal and therefore we get there a subsequence of A which corresponds to one of the two shapes given in Lemma 4.6.3. The slot between P and Q can be augmented at most three times because h0 (Q) = 3. Because of our general assumption that all blow-ups lie over the same fiber, we can even conclude by Lemma 4.6.2 that this slot even can be extended at most two times. For the same reason, if this slot has been extended two times, then the other slot neighbouring Q cannot be extended any more. Denote S1 the subset of the Ri used for augmenting the two slots around P . We have seen that S1 can consist of at most three elements. In the maximal case, we have S1 = {Ri1 , Ri2 , Ri3 } such that Ri1 ≤ Ri2 , Ri3 and Ri2 , Ri3 incomparable. In this case, the remaining two slots cannot

93

4.6. PROOF OF THEOREM 4.5.11

be augmented without violating our condition on (An )0 and thus the assertion follows. So we assume from now that S1 consists of at most two elements, which may be comparable or not. Also note that the base locus of P − Q coincides with the support of the total transform of b on X. Therefore, in the cases where either Ri1 and Ri2 are comparable, or S1 = {Ri1 } and Ri1 is used for augmentation in the slot between −P + Q and P , these divisors cannot come from blowing up points on or above b. If (An )0 = P , then the content of the two slots neighbouring this “bad” P must be of the form as given in Lemma 4.5.16 (ii). That is, we have a partition of the set of the Ri into three sets, S1 , S2 , S3 , where the latter two each consist of pairwise incomparable elements. If both S2 , S3 are empty, the assertion follows. If S1 consists of two elements, then only one of S2 , S3 can be nonempty, say S2 . If the two elements in S1 are incomparable, we have thus a partition into two subsets of incomparable elements and the assertion follows. If the two elements in S1 are comparable, i.e. Ri1 ≤ Ri2 , then we have (up to order inversion) the subsequence Ri1 , P − Ri1 − Ri2 , Ri2 in A. By Lemma 4.6.1 the divisor Ri1 must have multiplicity 1 with respect to P and Ri2 cannot come from blowing up a point on the fiber f . By this, after blowing down all Ri with Ei2 = −1 we are left with at most one chain of length 2, containing at least one Ei with Ei2 = −1 and we have f 2 = −1. So, by simultaneously blowing down these two divisors we arrive at some minimal surface X0 and the assertion follows. If S1 consists of only one element, then we have two possibilities. If Ri1 is used for augmentation in the slot between −P + Q and P , then the other slot neighbouring −P + Q is blocked for further augmentation and only one more slot is free for augmentation by incomparable Ri . So we can blow down X to X0 in at most two steps. If Ri1 is used for augmentation in the slot between P and Q, then we can have two nonempty sets S2 , S3 . Let us assume that the elements in S2 are used for augmentation between Q and P , and the elements in S3 for augmentation between −P + Q and P . As the base locus of −P + Q contains the support of the total transform of b on X, none of the Ri ∈ S3 are lying over any point of b. So, if Ri0 lies over some point in b, then it can be part of a chain of comparable Ri of length two. Hence, the maximal such length is at most two for all Ri . Hence the assertion follows. If Ri0 does not lie over some point of b, then the maximal length of a chain of comparable Ri which lie over some point of b is one, and after simultaneously blowing down the Ei with Ei2 = −1, there is no such Ri left. But then Ri0 might still be part of a chain of length 2, which will be the only such chain and another simultaneous blow-down will leave one of the components of this chain. However, now we can additionally blow down b instead and we will arrive at some other X0′ within two steps and the assertion follows. If (An )0 = 0, then An is located in one of the slots and the subsequence of A in this slot can be of one of the following forms: (13)

Q − P − Rj1 , Rj1 −

r X

Rjk , Rjr , Rjr−1 − Rjr , . . . , Rj2 − Rj3 , P − Rj1 − Rj2 − F,

k=2

(14) Q − Rj1 − Rj2 − G, Rj2 − Rj3 , . . . , Rjr−1 − Rjr , Rjr , Rj1 −

r X i=2

Rji −

s X

Rk i ,

i=1

Rks , Rks−1 − Rks , . . . , Rk1 − Rk2 , P − Rj1 − Rk1 − H, where F, G, H denote some possible additional summands coming from augmentations in the neighbouring slots. We denote S2 := {Rj2 , . . . , Rjr } and S2 := {Rk1 , . . . , Rks }. In the case 13, we have Rj1 ≤ Rji and the Rji incomparable for all i > 1. In the case 13, we have Rk1 ≤ Rji and the Rji incomparable for all i. If both S2 and S3 are empty, then A is an augmentation by

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the elements of S1 and by Rj1 and one possible augmentation by some Ri in the remaining slot. Then Ri and Rj1 must be comparable. These can form a chain of length at most three which cannot lie over b. Therefore we can conclude as before that we can blow-down the surface X to a surface X0′ in at most two steps. If S2 consists of two incomparable elements, then the other neighbouring slot of Q is blocked for augmentations and the remaining augmentation must be of the form (13) with Rj1 (and thus all the Rji ) not lying over b. So, if there exists a chain of length three, this chain cannot lie over b and again we can blow-down in two steps to some X0′ . If S1 consists of two comparable elements then the remaining augmentation must be of the form (13) where S2 = ∅, as G 6= 0. Then we possibly have a maximal chain of length four, where at least one of the elements in S1 and one of Rj1 and Rki involved have multiplicity one, and all the Rki incomparable. With similar arguments as before, we can always blowing down X to some X0′ in two steps by possibly contracting the fiber f . In the remaining cases we have to consider S1 consisting of one or two elements. The arguments are completely analogous to the previous arguments and we leave these to the reader.  We conclude that Theorem 4.5.11 follows from Propositions 4.6.5 and 4.6.6 in the case (An )0 6= −(a + s)P + Q. For the case (An )0 = −(a + s)P + Q we note that if fi 6= fj then Ri and Rj are incomparable. Moreover, from the proof of Proposition 4.6.4 we see that in order to blow-down to some X0′ we may have to blow-down the strict transform of some fiber. But any such choices can be made simultaneously. This proves Theorem 4.5.11. 4.7. Divisorial cohomology vanishing on toric surfaces Let X be a smooth complete toric surface whose associated fan is generated by lattice vectors l1 , . . . , ln and recall that Pic(X) is generated by the T -invariant divisors D1 , . . . , Dn . Recall from section 4.2 that, besides the coordinates associated to a minimal model X0 , we have two further coordinatizations for P Pic(X). The first is given by choosing for a given divisor D a T -invariant representative D ∼ ni=0 ci Di such that we can identify this representative n with a tuple P(c1 , . . . , cn ) in Z . The second coordinatization is given by tuples (d1 , . . . , dn ) ∈ Z such that i∈[n] di li = 0. The di are uniquely determined by the ci by the relations di = ci−1 + aiP ci + ci+1 for P i ∈ [n]. The ci are determined by the di up to a character m ∈ M , n n ′ ′ that is, i=0 ci Di iff there exists some m ∈ M such that ci = ci + li (m) for i=0 ci Di ∼ all i ∈ [n]. In what follows, we will use all of these coordinatizations for the classification of strongly left-orthogonal divisors on X. Pn Now assume that for a given divisor D, a T -invariant representative D ∼ i=0 ci Di is chosen. Then we can associate to D a hyperplane arrangement {Hi }i∈[n] in MQ which is given by hyperplanes Hi := {m ∈ MQ | li (m) = −ci }. The twist ci 7→ ci + li (m) for some m ∈ M then corresponds to a translation of the hyperplane arrangement by the lattice vector −m. The action of T induces an eigenspace decomposition of the space of global sections of O(D): M   H 0 X, O(D) m . H 0 X, O(D) ∼ = m∈M

 The nontrivial isotypical components H 0 X, O(D) m are one-dimensional and we have  H 0 X, O(D) m 6= 0 iff li (m) ≥ −ci for all i ∈ [n]

4.7. DIVISORIAL COHOMOLOGY VANISHING ON TORIC SURFACES

95

for m ∈ M , i.e. the non-vanishing isotypical components correspond to the characters which are contained in a distinguished chamber of the hyperplane arrangement. Pn Definition 4.7.1: Let D = i=1 ci Di be a torus invariant divisor, then we denote GD := {m ∈ M | H 0 (X, O(D))m 6= 0} = {m ∈ GD | li (m) ≥ −ci for all i ∈ [n]} and G◦D := {m ∈ GD | li (m) > −ci for all i ∈ [n]}. As the set GD counts the global sections of a T -invariant divisor D, by Serre duality, the set G◦D can naturally be associated with a T -eigenbasis of H 2 X, O(−D) . Namely, the canonical P divisor on X is given by KX = − ni=1 Di and h2 (−D) = h0 (KX + D) = |G◦D |. We want to interpret strong (pre-)left-orthogonality as a problem of counting lattice points, starting from GD for some strongly pre-left-orthogonal divisor D on P2 or Fa as classified in propositions 4.4.7 and 4.4.9. In general, the region containing GD is not quite a lattice polytope, but rather close to being one, as we will see in Proposition 4.7.12. This is illustrated in the following example. Example 4.7.2: Figure 4.1 shows examples for strongly pre-left-orthogonal divisors on P2 and Fa . The dots indicate the set GD , the white dots the subset G◦D . 3H: l2

l1

l3 l3

3Q − P: l2

l1

P: l4

Figure 4.1. The fans for P2 and F2 and the regions in M containing GD , for the cases D = 3H on P2 and D = P , D = 3Q − P on F2 , respectively. Consider any pre-left-orthogonal divisor βH, where β > 0, on P2 . Then it is easy to see that formulas (9) and (10),     β+2 β−1 χ(βH) = , χ(−βH) = , 2 2

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count GβH and G0βH , respectively. Similarly, formulas (11) and (12),     β+1 β χ(αP + βQ) = (α + 1)(β + 1) + a , χ(−αP − βQ) = (α − 1)(β − 1) + a , 2 2 count GαP +βQ and G◦αP +βQ , respectively. For the γi , there is a similar interpretation. Assume we have fixed a sequence of blowups b1 , . . . , bt as in the previous section, where every bk is toric. For some k ∈ [t], there are p, q, r ∈ [n] such that lp and lq span a cone in the fan of Xk−1 and lr = lp + lq represents the toric blow-up bk . We have: Lemma P j ∈ S span the fan of Xk−1 and denote P 4.7.3: Let p, q ∈ S ⊂ [n] such that the lj with D = i∈S ci Di a T -invariant divisor. Then b∗k D ∼ i∈S ci Di + (cp + cq )Dr and γi Ri ∼ cr Dr on Xk . Proof. Only the first assertion needs a proof. Let L the matrix whose rows are the li with i ∈ S and L′ the matrix consisting of the same rows as L but with the additional row lp + lq added between lp and lq . The assertion follows form the commutativity of the following diagram: 0

// M

0

// M

L

L′

//

//

Z|S| _

// Pic(Xi−1 ) _

// 0



 // Pic(Xi )

// 0.

Z|S|+1

 For given γk ≤ 0, we consider the lattice triangle which is inscribed by the lines Hp , Hq , Hr and whose lattice points we can count: Definition 4.7.4: Let lp , lq , lr be as before and γk ≤ 0, then we denote (i) Tγk := {m ∈ M | lp (m) ≥ −cp , lq (m) ≥ −cq , lr (m) ≤ −cr }, (ii) Tγ−k := {m ∈ M | lp (m) ≥ −cp , lq (m) ≥ −cq , lr (m) < −cr }, (iii) Tγ+k := {m ∈ M | lp (m) > −cp , lq (m) > −cq , lr (m) ≤ −cr }. As lp and lq form a basis of N , by translation by some m ∈ M we can assume without loss of generality that cp = cq = 0. Then, using Lemma 4.7.3, we can directly see that the  lattice points of Tγk , Tγ+k , Tγ−k are counted by binomial coefficients. We have |Tγk | = γk2−1 ,   |Tγ−k | = γ2k , and |Tγ+k | = γk2+1 . This is illustrated in the following example.

Example 4.7.5: With notation as before, figure 4.2 shows the local configuration  of lp , lq , lr and the relative positions of Hp , Hq , Hr for γk = −3. The dots indicate the −3−1 = 10 lattice 2   −3+1 + points in Tγk , the gray dots the = 3 lattice points in Tγk and the circled dots the −3 2 =6 2 lattice points in Tγ−k , with one lattice point in the intersection Tγ+k ∩ Ti− . P By Proposition 4.4.12, a pre-left-orthogonal divisor D is of the form (D)0 + ti=1 γi Ri with (D)0 pre-left-orthogonal on X0 and γi ≤ 0 for every i. The following proposition shows that strong pre-left-orthogonality is equivalent to that the Tγi cut out the lattice points of G(D)0 in a well-behaved manner.

4.7. DIVISORIAL COHOMOLOGY VANISHING ON TORIC SURFACES

97

Hk

l iq

li k

li p Hq

Hp

Figure 4.2. Three primitive vectors lp , lq , lr which pairwise generate N and the corresponding orthogonal hyperplane arrangement for γk = −3. Proposition 4.7.6: Let k > 0 and consider a blow-up bk : Xk −→ Xk−1 with notation as before. Let D be a pre-left-orthogonal divisor on Xk−1 and γk ≤ 0. Then b∗k D + γk Rk is preleft-orthogonal iff Tγ+k ⊂ G◦D . If D is strongly pre-left-orthogonal, then b∗k D + γk Rk is strongly pre-left-orthogonal iff Tγ+k ⊂ G◦D and Tγ−k ⊂ GD .   Proof. By Riemann-Roch we get χ(γk Rk ) = 1 − γ2k and χ(−γk Rk ) = 1 − γk2+1 . Moreover, we get χ(b∗k D + γk Rk ) = χ(D) + χ(γk Rk ) − 1 = χ(D) − γ2k and χ(−b∗k D − γk Rk ) =  χ(−D) + χ(−γk Rk ) − 1 = χ(−D) − γk2+1 . So we see that h1 (−b∗k D − γk Rk ) = 0 iff Tγ+k pre  cisely cuts γk2+1 lattice points out of G◦D and h1 (b∗k D + γk Rk ) = 0 iff Tγ+k precisely cuts γ2k lattice points out of GD and the assertion follows.  Consequently, we get: P Corollary 4.7.7: Let D = (D)0 + i γi Ri pre-left-orthogonal. Then D is left-orthogonal iff ` ` G(D)0 \ GD = tk=1 Tγ−k . Moreover, D is strongly left-orthogonal iff G(D)0 \ GD = tk=1 Tγ−k and ` G◦(D)0 = tk=1 Tγ+k .

In terms of lattice figures in M , strong left-orthogonality can be understood by proposition 4.7.6 and corollary 4.7.7 as follows. We start with an almost lattice polytope associated to a strongly pre-left-orthogonal divisor (D)0 on X0 and successively cut out lattice points of G(D)0 and G◦(D)0 by moving in hyperplanes Hr until G◦(D)0 +P γk Rk is empty and the sets {Tγ+k | k ∈ [t]} k and {Tγ−k | k ∈ [t]} form a “tiling” of G(D)0 \G(D)0 +Pk γk Rk and G◦(D)0 , respectively. We illustrate this in the following example.

Example 4.7.8: Figure 4.3 shows on the left the fan of F2 from figure 4.1 blown up three times by successively adding the primitive vectors l1 , l3 , and l2 . Note that the numbering of the Rj does not match with the numbering of the li , but rather the order in which the li were added to the fan. The right side shows the hyperplane arrangements for five examples of divisors D all of which have (D)0 = 3Q − P , with G3Q−P and G◦3Q−P shown in figure 4.1. In a) the hyperplanes H1 , H2 , H3 are indicated. The dark gray area indicates Tγ1 , the medium gray indicates Tγ2 , and the light gray Tγ3 . In a) we have D = 3Q − P − 2R1 ; here Tγ−1 cuts out three elements of G3Q−P and Tγ+1 cuts out one of G◦3Q−P . Therefore D is pre-left-orthogonal in this case. In b) we have D = 3Q − P − 2R1 and Tγ−3 cuts out only one of G3Q−P and Tγ+1 none of G◦3Q−P . Therefore D is

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CHAPTER 4. EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES

H3

H2

a

b

c

d

H1

l1

l3 l2

e Figure 4.3. The fan of F2 blown up three times and the hyperplane arrangements corresponding to the divisors a) 3Q − P − 2R1 , b) 3Q − P − 2R3 , c) 3Q − P − 2R1 − 2R2 , d) 3Q − P − 2R1 − R2 − 2R3 , e) 3Q − P − 2R1 − 2R2 − R3 .

not strongly pre-left-orthogonal. Note that R1 and R3 behave differently because li1 does form a basis of N with either of the two primitive vectors which belong to the fan of F2 and in whose positive span li1 is contained, whereas li3 does not. In the cases c), d), e), all Tγ−i and Tγ+i cut out the correct number of lattice points of G3Q−P and G◦3Q−P , respectively, such that precisely the two elements in G◦3Q−P are cut out. So in all these cases D is strongly left-orthogonal. We will also need to know how we can pass from the coordinates associated to a minimal model X0 to the di -coordinates. For this we first illustrate the correspondence between divisors P P of the form αP + βQ + ti=1 γi Ri and polygonal lines of the form i∈[n] di li = 0 in the following example. P Example 4.7.9: It is convenient to interpret the relation i∈[n] di li = 0 as closed polygonal lines. If we successively place the vectors di li end to end in NQ , we obtain a figure which can be viewed as a polygonal line complex embedded in the arrangement {Hi }i∈[n] , rotated by 90 degrees. Figure 4.4 shows examples of divisors on the surface shown in figure 4.3. Note that the order in which the di li are placed end to end is not canonical, but there are the two obvious choices (clockwise or counterclockwise) by which the line complex can be interpreted as being embedded in the corresponding hyperplane arrangement.

4.7. DIVISORIAL COHOMOLOGY VANISHING ON TORIC SURFACES

3Q − P P

99

3P − P − 2R 1− 2R2− R3

Figure 4.4. The fan of figure 4.3 and the polygonal lines associated to the divisors P , 3Q − P , and 3Q − P − 2R1 − 2R2 − R3 . The picture shows the hyperplane arrangements associated to these divisors rotated by 90 degrees and the polygonal lines embedded into them. To change from coordinates associated to X0 to di -coordinates, by linearity it suffices to  consider d1 (D), . . . , dn (D) , where D is one of P , Q, H, Ri , i ∈ [t]. For the following lemma we assume that the fan of X0 is generated by lb , lc , ld , le if X0 ∼ = Fa or by lb , lc , ld if X0 ∼ = P2 . In the first case we assume that lb + ld = alc . With respect to Rk , we choose lp , lq , lr as above. The following lemma is just an observation: Lemma 4.7.10: (i) If X0 ∼ = P2 , then di (H) = 1 if i ∈ {b, c, d} and di (H) = 0 otherwise. ∼ (ii) If X0 = Fa , then di (P ) = 1 if i ∈ {c, e} and di (P ) = 0 otherwise. (iii) If X0 ∼ = Fa , then dc (Q) = a, di (Q) = 1 if i ∈ {b, c}, and di (Q) = 0 otherwise. (iv) Without assumptions on X0 we have dp (Rk ) = dq (Rk ) = 1, dr (Rk ) = −1, and di (Rk ) = 0 otherwise. If we compare figure 4.4 with figure 4.3, we see that in these examples, for strongly leftorthogonal D, the associated polygonal line contains GD . More generally, we get: Lemma 4.7.11: Let D = (d1 , . . . , dn ) ∈ N1 (X) be a T -invariant curve on a smooth complete P toric surface X. If, as a divisor, D is numerically left-orthogonal, then χ(D) = i di .

P Proof. Let E = i∈[n] c′i Di , then it follows from the discussion in section 4.2 that D.E = P ′  i∈[n] di ci . We apply this to E = −KX = i∈[n] Di and use Lemma 4.3.3 (i).

P

P Of course, if D is strongly left-orthogonal, then it follows that h0 (D) = i di . If moreover, (D)0 is strongly pre-left-orthogonal, it follows by induction, starting from the classification of S propositions 4.4.7 and 4.4.8, that GD ⊂ di ≥0 Hk , i.e. the positive di attribute to the global k sections not only numerically, but the associated line segments bounding GD actually contain GD .

100 CHAPTER 4. EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES By Proposition 4.2.4 a divisor D is nef iff di ≥ 0 for every i. Then the associated polygonal line complex is the boundary of a lattice polytope in MQ . The figures of example 4.7.9 show that these strongly left-orthogonal divisors are almost nef, as in every case di ≥ −1 for every i ∈ [n]. This also holds in general: P Proposition 4.7.12: Let D be a strongly left-orthogonal divisor on X. Then i∈I di (D) ≥ −1 for every cyclic interval I ⊂ [n]. Proof. We choose some sequence of equivariant blow-downs to some minimal model X0 . Assume first that (D)0 = 0. Then by Lemma 4.4.11 D = Rk for some k or D = Rk − Rl for k 6= l ∈ [t] and Rk , Rl incomparable. For p, q, r as above, we have by Lemma 4.7.10 that di (Rk ) = −1 for i = r, di (Rk ) = −1 for i ∈ {p, q} and di (Rk ) = 0 else. So the assertion follows immediately for D = Rk . For D = Rk − Rl we have just to take into account that the Rk and Rl are incomparable. If (D)0 6= 0 we can assume without loss of generality that (D)0 is strongly pre-left-orthogonal. Otherwise, we have necessarily h0 (D) = h0 (D)0 ) = 0 and −(D)0 is strongly pre-left-orthogonal. Then if the statement is true for the case −(D)0 strongly pre-left-orthogonal, we have di ≤ 1 for every i and therefore by above discussion that −di ≥ −1 for every i. We show by induction on (D)k , k = 0, . . . , t that the assertion is true for a strongly preleft-orthogonal divisor D. For k = 0, the assertion is true by inspection of the classification of strongly pre-left-orthogonal divisors on P2 (proposition 4.4.7) and Fa (proposition 4.4.8). It also follows that dj = |G(D)0 ∩ Hi | if li belongs to fan associated to X0 , i.e. the di count the lattice length plus one of the bounding faces of the polygonal line inscribing G(D)0 . In the induction step we will show that this is still true for all triples p, q, r and all k > 0. For k > 0, let (D)k − (D)k−1 = γk Rk . Consider the triple lp , lq , lr as before, by Proposition 4.7.6 it is a necessary condition that Hp and Hq intersect in some m ∈ G(D)k \ G◦(D)k . Moreover, necessarily dp , dq ≥ −γk − 1 and the result follows from above characterization of di (Rk ).  Remark 4.7.13: If ai = −1 for some i, then we can find a basis of Pic(X) with respect to some minimal model X0 such that Rt = Di . For any strongly pre-left-orthogonal divisor D it follows that D = (D)t−1 + γt Rt for some γt ≤ 0. Therefore, we have di ≥ 0. If ai ≥ 0, the divisor Di necessarily is the strict transform of some torus invariant divisor on X0 . So by the classification 4.4.7 and 4.4.8, the only cases with ai ≥ 0 and di (D) = −1 is where X0 ∼ = P1 × P1 and D is the pullback of P − Q or Q − P . Otherwise, if ai > 0, then di ≥ 0. 4.8. Strongly exceptional sequences of invertible sheaves on toric surfaces The following results give a full classification of strongly exceptional sequences of invertible sheaves on smooth complete toric surfaces. Theorem 4.8.1: Let X be a smooth complete toric surface, then for every strongly exceptional toric system A there exists a sequence of blow-downs X = Xt → · · · → X0 , where X0 = P2 or X0 = Fa for some a ≥ 0 such that the normal form of A is a standard augmentation from X0 . As a corollary of Theorems 4.5.11 and 4.8.1 we thus obtain: Theorem 4.8.2: Let X 6= P2 be a smooth complete toric surface. Then there exists a full strongly exceptional sequence of invertible sheaves on X if and only if X can be obtained by equivariantly blowing up a Hirzebruch surface two times (in possibly several points in each step).

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We will prove Theorem 4.8.1 in the remaining sections. In this section we will state and prove some of its direct consequences. Corollary 4.8.3: Let X be a smooth complete toric surface. If there exists a strongly exceptional sequence of invertible sheaves on X, then rk Pic(X) ≤ 14. Proof. A Hirzebruch surface Fa has four torus fixed points. So, after blowing up some of these points, the resulting toric surface has up to 8 fixed points. After blowing up these, we get a toric surface X whose fan is generated by at most 16 lattice vectors and thus rk Pic(X) ≤ 14, and the statement follows from Theorem 4.8.2.  Example 4.8.4: Consider the toric surface which is given by the sequence of self-intersection numbers −2, −2, −1, −3, −2, 0, 1. It is easy to see that there is no way to blow-down this surface to any Hirzebruch surface in only two steps. So by Theorem 4.8.2 there does not exist a strongly exceptional sequence of invertible sheaves on this surface. This is the counterexample which has been verified by explicit computations in [HP06]. Now consider the blow-up of this surface given by −2, −2, −1, −3, −2, −1, −1, 0. This surface can be blown-down to a F1 in two steps by simultaneously blowing down two divisors in each step. Therefore by Theorem 4.5.9 there exist strongly exceptional sequences of invertible sheaves on this surface. More concretely, if the F1 is spanned by lattice vectors l1 , l2 , l3 , l6 with l3 = l2 + l6 , we subsequently add l7 = l1 + l6 , l8 = l1 + l7 , l4 = l3 + l6 and l5 = l4 + l6 . Then, for example, we get a family of strongly exceptional toric systems by R1 , R3 − R1 , P − R3 , sP + Q, P − R2 , R2 − R4 , R4 , −(s + 1)P + Q − R1 − R2 − R3 − R4 for s ≥ −1. For a cyclic strongly exceptional toric system A on X the associated toric surface Y (A) has a nef anti-canonical divisor. It turns out that this even is a necessary condition for X if X itself is a toric surface: Theorem 4.8.5: Let X be a smooth complete toric surface. If there exists a cyclic strongly exceptional sequence of invertible sheaves on X, then its anti-canonical divisor is nef. Proof. By Proposition 4.2.5 we have to show that ai ≥ −2 for every i. Assume that A = A1 , . . . , An is a cyclic strongly exceptional toric system assume that ai < −2 for some P and j j i. We denote di := di (Aj ) for every j ∈ [n]. Then j∈[n] di = ai + 2 < 0 by Proposition P 4.2.5. Because A is cyclic and strong, every sum j∈I Aj is strongly left-orthogonal for every P j proper cyclic interval I ⊂ [n]. In particular, j∈I di ≥ −1 for every such I by Proposition 4.7.12. Now assume that there exists j ∈ [n] such that dji = −1. Without ` loss of generality, we can assume that j = 1. Then by choosing a decomposition [n] \ {1} = I1 I2 , where I1 , I2 are intervals, we can consider A1 , A′1 , A′2 , a short toric system of length 3 as in example 4.2.14. Then d1i + di (A′1 ) ≥ −1 and d1i + di (A′2 ) ≥ −1, hence di (A′1 ) ≥ 0 and di (A′2 ) ≥ 0, and we get ai ≥ −3. Now assume that ai = −3. Then there exist at least two j such that dji = −1; P because otherwise, if there was only one j with dji = −1, the condition that nj=1 dji = −1 would imply that dki = 0 for all k 6= j and thus all the Ak with k 6= j are contained in a hyperplane in Pic(X), which is not possible. Let j, k such that dki , dji = −1. Then |k − j| > 1, as Al + Al+1 must be strongly left-orthogonal for every l ∈ [n]. So we can consider a short toric system to periodicity 4: A′1 , A′2 , A′3 , A′4 with di (A′1 ) = di (A′3 ) = −1. As A′1 + A′2 + A′3

102 CHAPTER 4. EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES and A′2 + A′3 + A′4 must be strongly left-orthogonal, this implies that di (A′2 ), di (A′4 ) ≥ 1 and so ai ≥ −2, a contradiction.  The converse is also true in the toric case: Theorem 4.8.6: If X is a smooth complete toric surface with nef anti-canonical divisor, then there exists a full cyclic strongly exceptional sequence of invertible sheaves on X. Proof. The case of P2 is clear, and Hirzebruch surfaces are covered by 4.5.2. For the remaining two del Pezzo surfaces the existence follows from Theorem 4.5.14. For the other cases, we give in table 2 a list of examples, one for each surface. By construction, these toric 5b 6b 6c 6d 7a 7b 8a 8b 8c 9

H − R1 , R1 , H − R1 − R2 , R2 , H − R2 H − R1 − R3 , R1 , H − R1 − R2 , R2 , H − R2 − R3 , R3 H − R1 − R3 , R1 , H − R1 − R2 , R2 , H − R2 − R3 , R3 P − R1 , R1 , Q − R1 − R2 , R2 , P − R2 , Q − P H − R1 − R2 , R2 , R1 − R2 , H − R1 − R3 − R4 , R4 , R3 − R4 , H − R3 -1, -2, 0, -1, -1, -2, -2 H − R1 − R3 , R3 , R1 − R3 , H − R1 − R2 − R4 , R4 , R2 − R4 , H − R2 -1, -2, -1, -2, -1, -2, -1, -2 P − R1 − R4 , R1 , Q − R1 − R2 , R2 , P − R2 − R3 , R3 , Q − R3 − R4 , R4 -1, -2, -1, -1, -2, -1, -2, -2 H − R1 − R2 − R4 , R4 , R2 − R4 , R1 − R2 , H − R1 − R3 , R3 − R5 , R5 , H − R3 − R5 -1, -2, -2, -2, -1, -2, 0, -2 P − R1 − R4 , R4 , R1 − R4 , P + Q − R1 − R3 , R3 − R2 , R2 , P − R2 − R3 , −P + Q -1, -2, -2, -1, -2, -2, -1, -2, -2 H − R1 − R4 − R5 , R4 , R1 − R4 , H − R1 − R3 − R6 , R6 , R3 − R6 , H − R2 − R3 − R5 , R2 , R5 − R2 Table 2. Cyclic strongly exceptional toric systems on toric surfaces with nef anti-canonical divisor. -1, -2, 0, 1, -1 -1, -2, -1, -1, 1, -1 -1, -2, 0, 0, -1, -2 -1, -2, -2, 0, 1, -2 -1, -1, -1, -1, -2, -1, -2

systems are exceptional and to check that these are indeed cyclic strongly exceptional is a direct application of Proposition 4.7.6 and Corollary 4.7.7. Note that for 8a and 8c we have given examples which are augmentations of cyclic strongly toric systems on P1 × P1 and there is an ambiguity of assigning P and Q. For 8a, both cases are cyclic strongly exceptional. For 8c, we choose Q to be the class of the unique torus invariant prime divisor with self-intersection zero on 8c.  4.9. Straightening of strongly left-orthogonal toric divisors In order to proof Theorem 4.8.1 we classify strongly left-orthogonal divisors on a given toric surface X. For this, we introduce in this section a procedure for simplifying a given strongly left-orthogonal divisor. We call this procedure a straightening. We will classify strongly leftorthogonal divisors up to straightening. Lemma 4.9.1: Let D be a T -invariant strongly left-orthogonal divisor on X and i ∈ [n] such that Di2 = −1. If di (D) < 0, then either h0 (D) = 0 or D = Di . Proof. We write D = γt Di + (D)t−1 , where Xt−1 is the blow-down of X along Di . Then di = −γt by Proposition 4.2.4 and Lemma 4.4.2. By Proposition 4.4.12 and Remark 4.4.13 this

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implies that (D)0 is not pre-left-orthogonal with respect to the choice of any minimal model X0 for X which factorizes through Xt−1 . But then we either have h0 (D) = 0 or (D)0 = 0 or both. If (D)0 = 0, then by Proposition 4.4.11 we have D = Rp − Rq for some p, q ∈ [t] or D = Rp for p ∈ [t]. In the first case, we also get h0 (D) = 0, in the second, we necessarily have Rp = Di by Lemma 4.7.10.  So for any strongly left-orthogonal divisor D which is not a prime divisor Dj , we will assume without loss of generality that di ≥ 0 for any i ∈ [n] such that Di2 = −1. Otherwise, we will just take −D instead of D. Let us write D = γt Di + (D)t−1 for X → Xt−1 the blow-down of Di . If −1 ≤ γt ≤ 0, then Tγ+t = ∅ and it follows from Lemma 4.4.6, Proposition 4.7.6, and Corollary 4.7.7 that (D)t−1 is strongly left-orthogonal on Xt−1 . By iterating, we obtain a sequence of blowdowns X = Xt → · · · → Xs , where s ≥ 0 and Xs lies over some (not necessarily completely P specified yet) minimal model X0 . We can write D = (D)s + ti=s+1 ǫi Ri , where ǫi ∈ {0, −1} for every i and Ri is the total transform on X of the exceptional divisor of the blow-up Xi → Xi−1 . The divisor (D)s now has the property that either (D)s coincides with a prime divisor Di on Xs with Di2 = −1 or di (D)s ≥ 2 with respect to every T -invariant prime  divisor Di on Xs 2 0 0 with Di = −1. It follows from Corollary 4.3.3 (iv) that h (D) = h (D)s + s − t.

Definition 4.9.2: Assume (D)s is constructed as above and does not coincide with a prime divisor Di on Xs . Then we call (D)s a straightening of D. A divisor D is straightened if D = (D)s (and consequently X = Xs ). In the sequel we will keep the index ‘s’ to denote that Xs has been chosen with respect to the straightening of some strongly left-orthogonal divisor. In general, s 6= 0 and a straightening (D)s is not unique. However, we will show that the existence of a straightened divisor imposes a strong condition on the geometry of X. Proposition 4.9.3: Let X be a smooth complete toric surface and D a straightened divisor on X. Then either −KX is nef or X ∼ = Fa with a ≥ 3. To prove Proposition 4.9.3 we first show an auxiliary statement. Let f ∈ [n] and denote e1 , . . . , er , g1 , . . . , gu ∈ [n] all indices i such that lf and li form a basis of N , where the enumeration is as follows. Consider the line generated by lf in NQ , Then all the ei are contained in one half plane bounded by this line and all the gj in the other. Moreover, we require that for any i < j, the vector lej is contained in the cone generated by lf and lei , and lgj is contained in the cone generated by lf and lgi , respectively. We denote S ⊂ [n] all i such that li is contained in one of theP cones σ1 , σ2 , where σ1 is generated by le1 and lf , and σ2 is generated by lg1 and lf . Let D = i∈[n] ci Di be a T -invariant divisor. We denote Zf := {m ∈ M | li (m) = −cf + 1},  Zf′ := m ∈ Z | li (m) > −ci for all i ∈ {f, e1 , . . . , ej , g1 , . . . , gk } ,  Zf′′ := m ∈ M | lf (m) = 0 and li (m) ≥ 0 for i ∈ {e1 , . . . , er , g1 , . . . , gu } .

Lemma 4.9.4: If af ≤ −3, then there exists m ∈ Zf such that li (m) > −ci for all i ∈ S. Proof. It follows from Proposition 4.2.3 that there exists a sequence of blow-downs X = Xt → · · · → Xp such that the cones generated by lf , le1 and lg1 do not contain any lattice vector which belongs to the fan associated to Xp . Correspondingly, we have injective maps φ : [r] → [t], ψ : [s] → [t] such that Rφi , Rψj are the total transforms the exceptional divisors associated to the primitive vectors lei and lgj , respectively. Then for i < j, we have Rφi < Rφj and Rψi < Rψj , respectively, and Rφi , Rψj incomparable for all i, j. Note that we have the relations

104 CHAPTER 4. EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES 0 = af lf +lej +lgk , where af = Df2 , and 0 = le1 +blf +lg1 for some b ≥ af . We write D = (D)p + Pt − + − + i=p+1 γi Ri . Then the Tγφi , Tγφi and Tγψi , Tγψi have to fulfill the conditions of Proposition P P 4.7.6 and Lemma 4.7.7. In particular, we have df = df (D) = ce1 +bcf +cg1 + ji=1 γφi + ki=1 γψi with df ≥ −1 by Proposition 4.7.12. Let ler (m) = −cer + ker and lgu (m) = −cgu + kgu for some m ∈ Zf and ker , kgu ∈ Z. Then we have ker + kgu = cer + af cf + cku − af = df − af . The number of solutions such that ker , kgu > 0 is given by max{0, df − af − 1 ≥ 1}, which is always nonzero for af ≤ −3. We denote . We claim that if af ≤ −3 then there exists m ∈ Zf′ such that li (m) > −ci for all i ∈ S. Assume there exists i ∈ S \ {f, e1 , . . . , er , g1 , . . . , gu } such that li (m) ≤ −ci for some m ∈ Zf′ . Without loss of generality, we assume that li is contained in σ1 . As li and lf do not form a basis of N , then the fact that the hyperplane Hi cuts out lattice points in T ′ implies that Hi also cuts out at least the same number of lattice points m of Zf′′ . But because af ≤ −3, we have |Zf′ | > |Zf′′ | and the claim follows.  Proof of Proposition 4.9.3. If there does not exist f ∈ [n] such that af < −2, then −KX is nef by Proposition 4.2.5. So if there exists such an f we show that Xs ∼ = Fa for a ≥ 3. With above notation there exists m ∈ M such that li (m) > −ci for all i ∈ S by Lemma 4.9.4. Assume first that there exists u ∈ [n] such that lu = −lf . In this case there do not exist lv which are contained in one of the cones generated by lu and le1 or lu and lg1 , respectively, because any blow-up of one of these cones would require a lattice vector li which forms a basis of N together with lu and therefore with lf . This lattice vector then would be one of the lei or lgj , which is excluded by assumption. But then the hyperplane Hu must pass through Zf , as otherwise h2 (−D) 6= 0, and (D)0 = kP + Q, where k ≥ −1, with respect to the minimal model X0 associated to the fan generated by le1 , lg1 , lf , lu . But G◦nP +Q = ∅ and thus γi ∈ {0, −1} for all p < i ≤ t and in fact γi = 0, as D is straightened. This implies X = X0 ∼ = F|b| , where le1 + blf + lg1 = 0. Such an lu necessarily exists in the following cases. If a > 1, then by the classification of toric surfaces lf must belong to any minimal model for X which can be obtained by blowing down Xp , and there necessarily exists lu = −lf . If a = 1, then le1 and lg1 form a basis of N and the blow-up of the cone generated by these two just yields lu . So either X0 = F1 or X0 = P2 . If a < −1, then none of le1 , lg1 , lf can be blown-down and thus together with −lf must span the fan of a minimal model F|b| . It remain to consider the cases b ∈ {0, −1} and there is no u ∈ [n] with lu = −lf . If a = 0, then le1 = −lg1 and le1 , lg1 , lf must be part of a fan of any minimal model X0 which is a blow-down of Xp . Moreover, there exists lv1 such that lf + blv1 + le1 = 0, where without loss of generality b > 0 (and therefore b > 1), and all li in the fan associated to Xp for i different from e1 , g1 , f , v1 , are contained in the cone generated by lv1 and lg1 . Then we have (D)0 = kP + lQ with respect to the coordinates in Pic(X0 ), where the fan of X0 is generated by le1 , lg1 , lf , lv1 . The divisor (D)0 is strongly pre-left-orthogonal and for any i ∈ / {e1 , g1 , f, v1 }, the index of the subgroup of N generated by lf and li is at least 3. Let v1 , . . . , vw ⊂ [n] P denote all elements such w forms a basis of N together with l and denote D = (D) + that l v g 0 1 i=2 γvi Ri + rest. Then Pw i γ ≤ k + 1 and because the index of the subgroup of N generated by lf and one of the v i i=2 ` + ∩ Z ′ = ∅, where η : {2, . . . , w} → [n] is the lvi with i > 1 is at least 3 and we have w T i=2 γηi f injective map which associates the Ri to the elements v2 , . . . , vw . Hence Zf′ must be empty and therefore af ≥ −2. In the last case, a = −1, for every i ∈ / {e1 , g1 , f } with li part of the fan associated to Xp , by our assumptions the index of the subgroup ` of N generated by lf and li is at least two and, similarly as in the previous case, we have i∈K Tγ+η ∩ Zf′ = ∅, where K ⊂ [n] denotes those i i such that li in the complement of σ1 and σ2 . Hence we have af ≥ −2. 

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Using Corollary 4.7.7 and Proposition 4.9.3 it is a rather straightforward exercise to go through table 1 and to find all possible straightened divisors. Proposition 4.9.5: Table 3 shows a complete list of straightened divisors and their associated toric surfaces. P2 111 H, 2H × P1 0000 P + sQ, Q + sP , where s ≥ −1 F1 0 -1 0 1 P , Q + sP , where s ≥ 1 F2 0 -2 0 2 P, 2Q − P, Q + sP , where s ≥ −1 Fa , a ≥ 3 0 -a 0 a P , Q + sP , where s ≥ −1 6d -1 -2 -2 0 1 -2 3H − 2R1 − R2 − R3 8a -1 -2 -1 -2 -1 -2 -1 -2 4H − 2(R1 + R2 + R3 ) − R4 − R5 8c -1 -2 -2 -2 -1 -2 0 -2 4H − 2(R1 + R2 + R4 ) − R3 − R5 9 -1 -2 -2 -1 -2 -2 -1 -2 -2 4H − 2(R1 + R3 + R5 ) − R2 − R4 − R6 Table 3. Classification of straightened divisors. The first column of the table shows the name of the surface as given in table 1, the second column shows the self-intersection numbers of the toric divisors, and the third columns lists the straightened divisors on the surface. The underlined intersection numbers indicate which divisors are blown-down to obtain a minimal model and the numbering of the Ri is just the left-to-right order of the underlined divisors. P1

It turns out that there exist only four straightened divisors which are realized on toric surfaces different from P2 or Fa . Their associated hyperplane arrangements and polygonal lines are shown in figure 4.5. 4.10. Proof of Theorem 4.8.1 Let A = A1 , . . . , An be a strongly exceptional toric P system on X. The first step for proving n−1 Theorem 4.8.1 is to consider the straightening of A := i=1 Ai and to find a preferred coordinate system for Pic(X) with respect to A. The idea here is that by Proposition 4.9.5 there are only the few possibilities for Xs listed in table 3, which are already close to a minimal model X0 . It follows from Proposition 4.10.2 that every strongly exceptional sequence on X is an augmentation of a sequence on Xs . In the case where Xs is the projective plane or a Hirzebruch surface, we have Xs = X0 and so by definition every augmentation of a strongly exceptional toric system on Xs is a standard augmentation. If If Xs is isomorphic to 6d, then the assertion of the theorem follows from Proposition 4.10.3. In remaining cases, i.e. Xs is one of 8a, 8c, 9, we show in Proposition 4.10.4 that X = Xs . These three cases are analyzed in Propositions 4.10.5, 4.10.6, and 4.10.7, which show that in every case A is a standard augmentation on Xs . This completes the proof of Theorem 4.8.1. Moreover, we draw the following corollary from Propositions 4.10.5, 4.10.6, and 4.10.7: Corollary 4.10.1: If Xs is one of 8a, 8c, 9, then X = Xs and A is cyclic. Now we prove the statements mentioned above. Proposition 4.10.2: Every strongly exceptional toric system has a normal form which is an augmentation of a strongly exceptional toric system on Xs .

106 CHAPTER 4. EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES 6d

8c

8a

9

Figure 4.5. The hyperplane arrangements and the polygonal lines associated to the four straightened divisors which are not realized on P2 or a Hirzebruch surface. The dots indicate the global sections. Pn−1 Proof. Let A = A1 , . . . , An be a strongly exceptional toric system and A := i=1 Ai and (A)s the straightening of A. We assume that X 6= Xs and denote Rt , . . . , Rs−1 the total transforms of the exceptional divisors of the blow-ups b1 , . . . , bs−1 and complete these to a basis of Pic(X) with respect to some X0 which is a blow-down of Xs . We may now assume that A is in normal form. The divisor Rt represents a torus invariant prime divisor of self-intersection −1 on X. Then A = (A)t−1 + γt Rt , where γt ∈ {0, −1}, and An = (An )t−1 + δt Rt , where γt + δt = −1. There must be at least two of the Ai which are not contained in the hyperplane Rt⊥ , as otherwise the projection (A1 )t−1 , . . . , (An )t−1 would also satisfy properties (i) and (ii) of Definition 4.2.6. But it is clear from the proof of Proposition 4.2.7 that this is not possible. So, as A is in normal form, there must be some Ai such that Ai = (Ai )t−1 + Rt and (Ai )0 = 0. Let i ∈ I = [i1 , i2 ] ⊂ [n − 1] be the maximal interval such that (Aj )0 = 0 for every j ∈ I. Then the sequence AI = Ai1 , . . . , Ai2 must be of one of the forms (i) or (ii)

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of Lemma 4.5.16. Moreover, there cannot be any other j ∈ [n] \ I such that (Aj )0 = 0 and Aj = (Aj )t−1 + Rt as this would necessarily contradict property (ii) of Definition 4.2.6. If AI is of the form of Lemma 4.5.16 (ii), we have two possibilities. First, Ai = Rt , which implies Ai−1 = (Ai−1 )t−1 − Rt (respectively An = (An )t−1 − Rt if i = 1) and Ai+1 = (Ai+1 )t−1 − Rt and (Aj )t−1 = 0 for every other j ∈ [n]. Therefore we can consider the projection (A1 )t−1 , . . . , (Ai−1 )t−1 , (Ai+1 )t−1 , . . . , (An )t−1 which is a strongly exceptional toric system in Pic(Xt−1 ). Second, Ai = Rt − Rk for some k < t and thus χ(Ai ) = 0, then, as in proposition 4.5.18, we can reorder the toric system by replacing Ai by −Ai , Ai−1 by Ai−1 + Ai and Ai+1 by Ai+1 + Ai , respectively, such that it remains strongly exceptional. In particular, we can reorder it such that Aj becomes Rt for some j ∈ I and apply the same argument as before. If AI is of the form of Lemma 4.5.16 (i), we can consider the divisors Ai1 −1 and Ai2 +1 , where we identify i1 − 1 with n if i1 = 1. Note that i2 − i1 < t, so that i1 − 1 6= i2 + 1. Now again by reordering, we can change A such that either Ai1 = (Ai1 )t−1 − Rt and Ai1 −1 = (Ai1 −1 )t−1 + Rt , or Ai2 = (Ai2 )t−1 − Rt and Ai2 +1 = (Ai2 +1 )t−1 + Rt . But then by our assumption on I and A being of normal form, one of i1 − 1, i2 + 1 must be equal to n. But above we have seen that δt ≤ 0, which is a contradiction, and AI cannot be of the form of Lemma 4.5.16 (i). Altogether we have seen now that A is an extension of a strongly exact toric system on Xt−1 and the proposition follows by induction.  Proposition 4.10.3: Let X be a toric surface isomorphic P to 6d and A = A1 , . . . , A6 a strongly exceptional toric system on X such that A = (A)s = 5i=1 Ai = 3H − 2R1 − R2 − R3 in the coordinates indicated in table 3. Then A is the augmentation of a standard sequence on X2 .

Proof. Clearly A6 = R1 , so A5 = (A5 )2 − R1Pand A1 = (A1 )2 − R1 . If we consider the projection (A1 )2 , . . . , (A5 )2 and denote AI := i∈I Ai for every interval I ⊂ [4], then (AI )2 = AI if 1 ∈ / I and (AI )2 − R1 = AI if 1 ∈ I and thus AI is strongly left-orthogonal for every such I and thus (A1 )2 , . . . , (A5 )2 is a strongly exceptional toric system on X2 and A an augmentation.  Denote P(A)s := {m ∈ MQ | li (m) ≥ −ci } the rational polytope containing G(A)s . Lemma 4.10.4: (i) Let X be a toric surface and A = A1 , . . . , An a strongly exceptional toric system on X such that A = (A)s and PAs has no corners in M . Then A cannot be augmented to a strongly exceptional sequence on any toric blow-up of X. (ii) In the cases where Xs is one of 8a, 8c, 9, the polytope P(A)s has no corners. P Proof. Write (A)s = ni=1 ci Di . From 4.7.6 it follows that for (A)s − Ri1 to be strongly left-orthogonal, there must exist a lattice point m ∈ GD and li , lj such that li (m) = −ci and lj (m) = −cj , i.e. m is a corner of P(A)s , and moreover, li1 must be contained in the positive span of li and lj . So it follows that (A)s cannot be a straightening of a divisor living on some blow-up of X of the form (A)s − Ri1 − · · · − Rik , where i1 , . . . , ik > t. Now consider A′ = A′1 , . . . , An+k a toric system which is an augmentation of A. As (A)s = (A′ )s′ , where s′ = s + k, the augmentation process can only happen between An−1 and An , or between An P and A1 . But then there exists n′ > l > n − 1 such that li=1 A′i = As − Ril with il > t, which cannot be strongly left-orthogonal, which proves (i). For (ii) we refer to figure 4.5.  We observe that the condition of lemma 4.10.4 are fulfilled for the remaining three cases. Proposition 4.10.5: Let X be a toric surface isomorphic P7 to 8a and A = A1 , . . . , A8 a strongly exceptional toric system on X such that A = (A)s = i=1 Ai = 4H − 2(R1 + R2 + R3 ) − R4 − R5

108 CHAPTER 4. EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES in the coordinates indicated in table 3. Then A is cyclic strongly exceptional and its normal form is an extension of the standard toric system on P2 . Without bringing it into normal form, the toric system cannot be extended to a strongly exceptional toric system on any toric blow-up of X. Proof. The latter assertion follows by Lemma 4.10.4. To prove the first Pclaim, we have to check that for any nonempty cyclic interval ∅ = 6 I ( [8] the divisor AI := i∈I Ai is strongly left-orthogonal. By assumption, this is true for every I which does not contain n, and it thus remains to check the complementary intervals [n] \ I for n ∈ / I. For A8 = −H + R1 + R2 + R3 2 = 4 it follows that χ(A ) ≤ 4 for every ∅ 6= I ( [8] by we have χ(A8 ) = 0 and with KX I Lemma 4.3.3 (iii). Using Proposition 4.7.6 and Corollary 4.7.7 together with formulas (9) and (10), it is a straightforward exercise to determine all strongly left-orthogonal divisors with Euler characteristic at most 4. These are shown in table 4. We see that almost all elements in this table χ(D) 0 1

2 3 4

D Ri − Rj with {i, j} = 6 {1, 5}, {3, 4}, ±(H − Ri − Rj − Rk ) with i, j, k pairwise distinct and {i, j, k} = 6 {1, 2, 5}, {2, 3, 4} Ri for i ∈ {1, 2, 3, 4, 5}, H − Ri − Rj for i 6= j, 2H − R1 − R2 − R3 − R4 − R5 , H− PRi for i ∈ {1, 2, 3, 4, 5}, 2H − i6=j Rj for i ∈ {1, 2, 3, 4, 5} H, 2H − Ri − Rj − Rk P with i, j, k pairwise distinct, 3H − 2Ri − j6=i Rj for any i P 2H − Ri − Rj for i 6= j 3H − 2Ri − j6=i,k Rj for k 6= i and (i, k) 6= (1, 5), (3, 4), 4H − 2(Ri + Rj + Rk ) − Rl − Rm for i, j, k, l, m pairwise distinct, 5H − 3Ri − 2(Rj + Rk + Rl ) − Rm for i, j, k, l, m pairwise distinct and i ∈ {1, 4, 5} Table 4. Strongly left-orthogonal divisors with Euler characteristic ≤ 4 on the variety 8a

P can be paired, i.e. if some D is in the table, then also −KX −D is. So, because −KX = 8i=1 Ai , it follows that if AI is in the table, then A[n]\I is and the proposition follows. The only exceptions which cannot be completed to a strongly left-orthogonal pair are 2H − R3 − R4 , 2H − R1 − R5 , 3H − R2 − R3 − R4 − 2R5 , 3H − R1 − R2 − 2R4 − R5 , 4H − 2(R1 + R2 + R5 ) − R3 − R4 , 4H − 2(R2 + R3 + R4 ) − R1 − R5 , and 5H − 3Ri − 2(Rj + Rk + Rl ) − Rm . We show that these cannot be of the form AI for I ⊂ [n − 1]. The case 5H+ rest can be excluded at once, as by assumption A is in normal form with respect to X0 , hence we always have (AI )0 = βH with β < 4. With respect to A and I = [k, l] with 1 ≤ k < l < n, we consider the following four divisors: C1 , AI , C2 , A8 , where AI as before Pk−1 Pn−1 and A8 = −H + R1 + R2 + R3 as before, and C1 := j=1 Aj , C2 := j=l+1 Aj , where C1 = 0 if k = 1 and C2 = 0 if l = n − 1. Because of the properties of toric systems, we have that A8 .(C1 + C2 ) = AI .(C1 + C2 ) ∈ {0, 1, 2}, depending on the Ci being nonzero or not. Now let us assume that AI = 2H − R3 − R4 . Then C1 + C2 = −KX − A8 − AI = 2H − R1 − 2(R2 + R3 ) − R5 and A8 .(C1 + C2 ) = 3, which is not possible. If AI = 3H − R2 − R3 − R4 − 2R5 , we get C1 + C2 = H + R5 − 2R1 − R2 − R3 and (C1 + C2 ).A8 = 3. Therefore this case is also excluded.

4.10. PROOF OF THEOREM 4.8.1

109

If AI = 4H − 2(R1 + R2 + R5 ) − R3 − R5 ), then (C1 + C2 ) = R5 − R3 and A8 .(C1 + C2 ) = −1, which is not possible. The remaining three cases differ only by enumeration from the first three and can be excluded analogously. Altogether, under the conditions of the proposition, the strongly exceptional toric system A is always cyclic. If we bring it into normal form by inverting A8 , we get that A′ = 2H − R4 − R5 and (A′ )s = 2H. So by Proposition 4.10.2 and the subsequent remark, the toric system is an extension of the toric system H, H, H on P2 .  Proposition 4.10.6: Let X be a toric surface isomorphic to 8c and A = A1 , . . . , A8 a strongly P exceptional toric system on X such that A = (A)s = 7i=1 Ai = 4H − 2(R1 + R2 + R4 ) − R3 − R5 in the coordinates indicated in table 3. Then A is cyclic strongly exceptional and its normal form is an extension of the standard toric system on P2 . Without bringing it into normal form, the toric system cannot be extended to a strongly exceptional toric system on any toric blow-up of X. Proof. In this case the arguments are completely analogous to the proof of proposition 4.10.5. The only difference is the classification of strongly left-orthogonal divisors with Euler characteristic at most four, which is shown in table 5. In table 6 we list the divisors D from table χ(D) 0 1

2 3 4

D ±(Ri − Rj ) with i ∈ {1, 2, 3}, j ∈ {4, 5}, ±(H − Ri − Rj − Rk ) with i 6= j ∈ {1, 2, 3}, k ∈ {4, 5} Ri for any i, H − Ri − Rj for i 6= j, 2H − R1 − R2 − R3 − R4 − R5 , H− PRi for any i, 2H − i6=j Rj for any i H, 2H − Ri − Rj − Rk P with i, j, k pairwise distinct, 3H − 2Ri − j6=i Rj for any i 2H − Ri − Rj for i 6= j P 3H − 2Ri − j6=i,k Rj for k 6= i and (i, k) 6= (4, 5), (2, 3), (1, 3), (1, 2), 4H − 2(Ri + Rj + Rk ) − Rl − Rm for i, j, k, l, m pairwise distinct, i, j ∈ {1, 2, 3}, k ∈ {4, 5}, 5H − 3Ri − 2(Rj + Rk + Rl ) − Rm for i, j, k, l, m pairwise distinct and i ∈ {4, 5} Table 5. Strongly left-orthogonal divisors with Euler characteristic ≤ 4 on the variety 8c.

5 which are candidates for some AI and do not have a strongly left-orthogonal partner together with C := A − D, and the intersection numbers C.D, C.A8 . As we can see, we get in every case that the intersection numbers are not compatible with AI coming of a toric system. So, under the conditions of the proposition, the strongly exceptional toric system A is always cyclic. If we bring it into normal form by inverting A8 , we get that A′ = 2H − R3 − R5 and (A′ )s = 2H. So by Proposition 4.10.2 and the subsequent remark, the toric system is an extension of the toric system H, H, H on P2 .  Proposition 4.10.7: Let X be a toric surface isomorphic P7to 9 and A = A1 , . . . , A9 a strongly exceptional toric system on X such that A = (A)s = i=1 Ai = 4H − 2(R1 + R3 + R5 ) − R2 − R4 − R6 in the coordinates indicated in table 3. Then A is cyclic strongly exceptional and

110 CHAPTER 4. EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES D C C.D C.A8 2H − R4 − R5 2H − 2(R1 + R2 ) − R3 − R4 3 3 3H − 2R5 − R1 − R2 − R3 H + R5 − R1 − R2 − 2R4 3 0 3H − 2R3 − R1 − R4 − R5 H + R3 − R1 − 2R2 − R4 3 2 3H − 2R3 − R2 − R4 − R5 H + R3 − 2R1 − R2 − R4 3 2 3H − 2R2 − R3 − R4 − R5 H − 2R1 − R4 2 1 Table 6. Testing intersection numbers of some divisors of table 5. its normal form is an extension of the standard toric system on P2 . Without bringing it into normal form, the toric system cannot be extended to a strongly exceptional toric system on any toric blow-up of X. Proof. The proof is analogous to propositions 4.10.5 and 4.10.6. Here, we have χ(A) = 3, and table 7 shows the strongly left-orthogonal divisors with Euler characteristic ≤ 3. The χ(D) 0

D Ri − Rj with {i, j} = 6 {1, 2}, {3, 4}, {5, 6}, ±(H − Ri − Rj − Rk ) with i, j, k pairwise distinct, {i, j, k} \ {1, 2} = 6 {5}, {6}; {i, j, k} \ {3, 4} = 6 {1}, {2}; {i, j, k} \ {5, 6} = 6 {3}, {4}, 2H − R1 − R2 − R3 − R4 − R5 − R6 1 Ri for any i, H −R Pi − Rj for i 6= j, 2H − j6=i Rj for any i, 2 H − Ri for any i, P 2H − k6=i,j Rk for any i 6= j, P 3H − 2Ri − j6=i Rj for any i 3 H, 2H −P Ri − Rj − Rk with i, j, k pairwise distinct, 3H − 2Ri − k6=i,j Rj for any i 6= j and j 6= i + 1 if i odd, 4H − 2(Ri + Rj + Rk ) − Rl − Rm − Rn with i, j, k, l, m, n pairwise distinct, {i, j, k} \ {1, 2} = 6 {5}, {6}; {i, j, k} \ {3, 4} = 6 {1}, {2}; {i, j, k} \ {5, 6} = 6 {3}, {4}, 5H − 2(R1 + R2 + R3 + R4 + R5 + R6 ) Table 7. Strongly left-orthogonal divisors with Euler characteristic ≤ 3 on the variety 9

unpaired divisor 5H − 2(R1 + R2 + R3 + R4 + R5 + R6 ) can be excluded as once, as A is in normal form. For the other cases, we make use of the Z3 -symmetry of the table and consider only three cases, and the others follow the same way by exchanging indices. Assume first AI = 3H − 2R2 − R3 − R4 − R5 − R6 , then C := A − AI = H − 2R1 − R3 − R5 . Then C.A9 = C.(−H + R1 + R3 + R5 ) = 3, which is not possible. The next case is AI = (2H − R1 − R2 − R5 ). Then C = 2H − R1 − 2R3 − R5 − R6 and C.A9 = −1, which is not possible. The last case is AI = 4H − 2(R1 + R2 + R5 ) − R3 − R4 − R6 . Then C = R2 − R3 and C.A9 = −1, and this case also is excluded. Again, altogether we get that under the conditions of the proposition, the strongly exceptional toric system A is always cyclic. If we bring it into normal form by inverting A9 , we get that A′ = 2H − R2 − R4 − R6 and (A′ )s = 2H. So by Proposition 4.10.2 and the subsequent remark, the toric system is an extension of the toric system H, H, H on P2 . 

Bibliography [AH99]

K. Altmann and L. Hille. Strong exceptional sequences provided by quivers. Alg. Represent. Theory, 2(1):1–17, 1999. [AKO06] D. Auroux, L. Katzarkov, and D. Orlov. Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves. Invent. Math., 166(3):537–582, 2006. [AKO08] D. Auroux, L. Katzarkov, and D. Orlov. Mirror symmetry for weighted projective planes and their noncommutative deformations. Ann. of Math., 167(3):867–943, 2008. [AM04] P. S. Aspinwall and I. V. Melnikov. D-branes on vanishing del Pezzo surfaces. JHEP, 1204:42, 2004. [Asp08] P. S: Aspinwall. D-Branes on Toric Calabi-Yau Varieties. arXiv:math/0806.2612 , 2008. [Ath99] A. Athanasiadis. The largest intersection lattice of a discriminantal arrangement. Beitr. Algebra Geom., 40(2):283–289, 1999. [AV85] M. Artin and J.-L. Verdier. Reflexive modules over rational double points. Math. Ann., 270(1):79–82, 1985. [Bae88] D. Baer. Tilting sheaves in representation theory of algebras. Manuscripta Math., 60:323–347, 1988. [BB97] M. Bayer and K. A. Brandt. Discriminantal arrangements, fiber polytopes and formality. J. Algebraic Combin., 6(3):229–246, 1997. [Be˘ı78] A. A. Be˘ılinson. Coherent sheaves on Pn and problems of linear algebra. Funct. Anal. Appl., 12(3):214– 216, 1978. [BFS90] L. J. Billera, P. Filliman, and B. Sturmfels. Constructions and Complexity of Secondary Polytopes. Advances in Mathematics, 83:155–179, 1990. [BG02] W. Bruns and J. Gubeladze. Semigroup algebras and discrete geometry. In S´eminaires et Congr`es, volume 6, pages 43–127. Soc. Math. France, Paris, 2002. [BG03] W. Bruns and J. Gubeladze. Divisorial Linear Algebra of Normal Semigroup Rings. Algebras and Representation Theory, 6:139–168, 2003. [BGG78] I. N. Bernstein, I. M. Gel’fand, and S. I. Gel’fand. Algebraic bundles on Pn and problems of linear algebra. Funct. Anal. Appl., 12(3):212–214, 1978. [BGS87] R.-O. Buchweitz, G.-M. Greuel, and F.-O. Schreyer. Cohen-Macaulay modules on hypersurface singularities II. Invent. Math., 88:165–182, 1987. [BH08] L. Borisov and Z. Hua. On the conjecture of King for smooth toric Deligne-Mumford stacks. Preprint, 2008. arXiv:0801.2812. [BLS+ 93] A. Bj¨ orner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler. Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1993. [Bon90] A. I. Bondal. Representation of associative algebras and coherent sheaves. Math. USSR Izvestiya, 34(1):23–42, 1990. [Bon00] L. Bonavero. Sur des vari´et´es toriques non projectives. Bull. Soc. Math. Fr., 128(3):407–431, 2000. [BP94] A. I. Bondal and A. E. Polishchuk. Homological properties of associative algebras: the method of helices. Russ. Acad. Sci., Izv. Math., 42(2):219–260, 1994. [BP06] A. Bergman and N. Proudfoot. Moduli spaces for D-branes at the tip of a cone. JHEP, 0603:73, 2006. [BP08] A. Bergman and N. Proudfoot. Moduli spaces for Bondal quivers. Pacific J. Math., 237(2):201–221, 2008. [BR86] M. Beltrametti and L. Robbiano. Introduction to the theory of weighted projective spaces. Expo. Math., 4:111–162, 1986. [Bri05] T. Bridgeland. t-structures on some local Calabi–Yau varieties. J. Alg., 289(2):453–483, 2005. [Bri06] T. Bridgeland. Derived categories of coherent sheaves. In Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures, pages 563–582. Z¨ urich: European Mathematical Society (EMS), 2006. [Bro06] N. Broomhead. Cohomology of line bundles on a toric variety and constructible sheaves on its polytope. arXiv:math/0611469, 2006. 111

112 [BV97] [Cas03] [CM04] [CM05] [Coh86] [Cox95] [Cra84] [CS05] [CS06] [CT00] [CV03] [Dan78] [Del75] [Dem80] [DL85] [Dol82] [Dou01] [Eik92] [EK85] [EMS00] [EW91] [Fal94] [Fuj03] [Fuj07] [Ful93] [GKZ94] [Gor89] [Gro68]

[GSV83] [GW78] [Hap88] [Har66] [Har77] [HHK07] [HHV06] [Hil04]

CHAPTER 4. BIBLIOGRAPHY M. Brion and M. Vergne. Residue formulae, vector partition functions and lattice points in rational polytopes. J. Am. Math. Soc., 10(4):797–833, 1997. C. Casagrande. Contractible classes in toric varieties. Math. Z., 243(1):99–126, 2003. L. Costa and R. M. Mir´ o-Roig. Tilting sheaves on toric varieties. Math. Z., 248:849–865, 2004. L. Costa and R. M. Mir´ o-Roig. Derived categories of projective bundles. Proc. Amer. Math. Soc., 133(9):2533–2537, 2005. D. C. Cohen. A Conjecture about Compact Quotients by Tori. In Arrangements — Tokyo 1998, volume 27 of Adv. Studies in Pure Mathematics, pages 59–68. Math. Soc. Japan, 1986. D. A. Cox. The Homogeneous Coordinate Ring of a Toric Variety. J. Algebr. Geom. 4, 1:17–50, 1995. H. Crapo. Concurrence geometries. Adv. Math., 54:278–301, 1984. A. Craw and G. Smith. Toric varieties are fine moduli spaces of quiver representations. preprint, 2005. A. Craw and G. G. Smith. Projective toric varieties as fine moduli spaces of quiver representations. arXiv:math/0608183, 2006. B. Chen and V. Turaev. Counting lattice points of rational polyhedra. Adv. in Math., 155:84–97, 2000. D. C. Cohen and A. N. Varchenko. Resonant local systems on complements of discriminantal arrangements and sl2 representations. Geom. Dedicata, 101:217–233, 2003. V. I. Danilov. Geometry of toric varieties. Russ. Math. Surv., 33(2):97–154, 1978. translation from Usp. Mat. Nauk 33, No. 2(200). C. Delorme. Espaces projectifs anisotropes. Bull. Soc. Math. France, 103:203–223, 1975. M. Demazure. Surfaces de Del Pezzo. I. II. III. IV. V. In Semin. sur les singularites des surfaces, Cent. Math. Ec. Polytech., Palaiseau 1976-77, volume 777 of Lect. notes Math., pages 21–69. Springer, 1980. ´ Norm. J. M. Drezet and J. Le Potier. Fibr´es stables et fibr´es exceptionnels sur P2 . Ann. scient. Ec. Sup. 4e s´erie, 18:193–244, 1985. I. Dolgachev. Weighted projective varieties. In Group actions and vector fields, Proc. Pol.-North Am. Semin., Vancouver 1981, pages 34–71, 1982. M. R. Douglas. D-branes, Categories and N=1 Supersymmetry. J. Math. Phys., 42:2818–2843, 2001. M. Eikelberg. The Picard group of a compact toric variety. Result. Math., 22(1–2):509–527, 1992. H. Esnault and H. Kn¨ orrer. Reflexive modules over rational double points. Math. Ann., 272(4):545–548, 1985. D. Eisenbud, M. Mustat¸ˇ a, and M. Stillman. Cohomology on Toric Varieties and Local Cohomology with Monomial Supports. J. Symb. Comput., 29(4–5):583–600, 2000. G. Ewald and U. Wessels. On the ampleness of invertible sheaves in complete projective toric varieties. Results Math., 19(3–4):275–278, 1991. M. Falk. A note on discriminantal arrangements. Proc. AMS, 122(4):1221–1227, 1994. O. Fujino. Notes on toric varieties from Mori theoretic viewpoint. Tohoku Math. J., 55(3):551–564, 2003. O. Fujino. Multiplication maps and vanishing theorems for toric varieties. Math. Z., 257(3):631–641, 2007. W. Fulton. Introduction to Toric Varieties. Princeton University Press, 1993. I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, resultants and multidimensional determinants. Math. Theory Appl. Birkh¨ auser, 1994. A. L. Gorodentsev. Exceptional bundles on surfaces with a moving anticanonical class. Math. USSR Izvestiya, 33(1):67–83, 1989. A. Grothendieck. Cohomologie locale des faisceaux coh´erents et th´eor`emes de lefschetz locaux et globaux (SGA 2), volume 2 of Advanced Studies in Pure Mathematics. North-Holland Publishing Co., Amster´ dam; Masson & Cie, Editeur, Paris, 1968. G. Gonzalez-Sprinberg and J.-L. Verdier. Construction g´eom´etrique de la correspondance de McKay. Ann. Sci. Ecole Norm. Sup., 16:409–449, 1983. S. Goto and K. Watanabe. On graded rings. II: Zn -graded rings. Tokyo J. Math., 1:237–261, 1978. D. Happel. Triangulated categories in the representation theory of finite-dimensional algebras, volume 119 of London Mathematical Society lecture note series. Cambridge University Press, 1988. R. Hartshorne. Residues and Duality, volume 20 of Lecture Notes in Mathematics. Springer, 1966. R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics. Springer, 1977. L. A. H¨ ugel, D. Happel, and H. Krause, editors. Handbook of tilting theory, volume 332 of London Mathematical Society lecture note series. Cambridge University Press, 2007. A. Hanany, C. P. Herzog, and B. Vegh. Brane Tilings and Exceptional Collections. JHEP, 0607:1, 2006. L. Hille. Exceptional Sequences of Line Bundles on Toric Varieties. In Y. Tschinkel, editor, Mathematisches Institut Universit¨ at G¨ ottingen, Seminars WS03-04, pages 175–190, 2004.

113 [HK06]

C. P. Herzog and R. L. Karp. Exceptional collections and D-branes probing toric singularities. JHEP, 0608:61, 2006. [HKP05] M. Hering, A. K¨ uronya, and S. Payne. Asymptotic cohomological functions of toric divisors. Preprint, to appear in Adv. Math., 2005. math.AG/0501041. [HP06] L. Hille and M. Perling. A Counterexample to King’s Conjecture. Comp. Math, 142(6):1507–1521, 2006. [Huy06] D. Huybrechts. Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, 2006. [HvdB07] L. Hille and M. van den Bergh. Fourier-Mukai transforms. In Handbook of tilting theory, volume 332 of London Math. Soc. Lecture Note Ser., pages 147–177. Cambridge Univ. Press, 2007. [Kap85] M. M. Kapranov. On the derived category of coherent sheaves on Grassmann manifolds. Math USSR Izvestiya, 24:183–192, 1985. [Kap86] M. M. Kapranov. Derived category of coherent bundles on a quadric. Funct. Anal. Appl., 20(2):141–142, 1986. [Kap88] M. M. Kapranov. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. math., 92(3):479–508, 1988. [Kaw06] Y. Kawamata. Derived categories of toric varieties. Michigan Math. J., 54(3):517–535, 2006. [Kaw08] Y. Kawamata. Derived Categories and Birational Geometry. Preprint, 2008. arXiv:0804.3150. [Kin97] A. King. Tilting bundles on some rational surfaces. Unpublished manuscript, 1997. [KN98] B. V. Karpov and D. Yu. Nogin. Three-block exceptional sets on del Pezzo surfaces. Izv. Math., 62(3):429–463, 1998. [KO95] S. A. Kuleshov and D. O. Orlov. Exceptional sheaves on del Pezzo surfaces. Russ. Acad. Sci., Izv. Math., 44(3):337–375, 1995. [Kon95] M. Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich 1994), pages 120–139. Birkh¨ auser, 1995. [Kon98] M. Kontsevich. Lectures at ENS, Paris, Spring 1998, notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet and H. Radriombololona. 1998. [Kul97] S. A. Kuleshov. Exceptional and rigid sheaves on surfaces with anticanonical class without base components. J. Math. Sci., 86(5):2951–3003, 1997. [Kuz05] A. Kuznetsov. Exceptional collections for Grassmannians of isotropic lines. Preprint, 2005. arXiv:math/0512013. [Man86] Y. I. Manin. Cubic forms. Algebra, geometry, arithmetic, volume 4 of North-Holland Mathematical Library. North-Holland, 2nd edition, 1986. [Mat02a] E. N. Materov. The Bott formula for toric varieties. Mosc. Math. J., 2(1):161–182, 2002. [Mat02b] K. Matsuki. Introduction to the Mori Program. Universitext. Springer, 2002. [MO78] K. Miyake and T. Oda. Lectures on Torus Embeddings and Applications. Springer, 1978. [MS89] Y. I. Manin and V. V. Schechtman. Arrangements of hyperplanes, higher braid groups and higher Bruhat orders. In J. Coates et al., editors, Algebraic number theory. Papers in honor of K. Iwasawa on the occasion of his 70th birthday on September 11, 1987, volume 17 of Advances Studies in Pure Mathematics, pages 67–84. Braunschweig: Vieweg, 1989. [MS04] E. Miller and B. Sturmfels. Combinatorial Commutative Algebra, volume 227 of Graduate Texts in Mathematics. Springer, 2004. [Mus02] M. Mustat¸ˇ a. Vanishing Theorems on Toric Varieties. Tohoku Math. J., II. Ser., 54(3):451–470, 2002. [Nog94] D. Y. Nogin. Helices on some Fano threefolds: constructivity of semiorthogonal bases of K0 . Ann. Sci. Ecole Norm. Sup., 27:129–172, 1994. [Oda88] T. Oda. Convex Bodies and Algebraic Geometry, volume 15 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, 1988. [OP91] T. Oda and H. S. Park. Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decomposition. Tohoku Math. J., II Ser., 43(3):375–399, 1991. [Orl93] D. O. Orlov. Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Russian Acad. Sci. Ivz. Math., 41(1):133–141, 1993. [OSS80] C. Okonek, M. Schneider, and H. Spindler. Vector bundles on complex projective spaces, volume 3 of Progress in Mathematics. Birkh¨ auser, 1980. [Pay04] S. Payne. Fujita’s very ampleness conjecture for singular toric varieties. Tohoku Math. J. (2), 58(3):447– 459, 2004. [Per04a] M. Perling. Graded Rings and Equivariant Sheaves on Toric Varieties. Mathematische Nachrichten, 263–264:181–197, 2004.

114

CHAPTER 4. BIBLIOGRAPHY

[Per04b] M. Perling. TiltingSheaves — a program to compute strongly exceptional collections on toric varieties. http://www.mathematik.uni-kl.de/~perling/ts/ts.html, 2004. [Per07] M. Perling. Divisorial cohomology vanishing on toric varieties. submitted, 2007. arXiv:0711.4836. [Ram05] J. L. Ram´ırez Alfons´ın. The Diophantine Frobenius Problem, volume 30 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2005. [Rei83] M. Reid. Decomposition of toric morphisms. In M. Artin and J. Tate, editor, Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, volume 36 of Prog. Math., pages 395–418. Birkh¨ auser, 1983. [Rei99] V. Reiner. The generalized Baues problem. In L. J. Billera et al., editors, New perspectives in algebraic combinatorics, pages 293–336. Cambridge University Press, 1999. [Rud90] A. N. Rudakov. Helices and vector bundles: seminaire Rudakov, volume 148 of London Mathematical Society lecture note series. Cambridge University Press, 1990. [Sam05] A. Samokhin. On the derived category of coherent sheaves on a 5-dimensional Fano variety. C. R. Math. Acad. Sci. Paris, 340(12):889–893, 2005. [Sam07] A. Samokhin. Some remarks on the derived categories of coherent sheaves on homogeneous space. J. Lond. Math. Soc., 76(2):122–134, 2007. [Sat00] H. Sato. Toward the classification of higher-dimensional Fano varieties. Tohoku Math. J., 52:383–413, 2000. [Sta82] R. P. Stanley. Linear diophantine equations and local cohomology. Invent. Math., 68:175–193, 1982. [Sta96] R. P. Stanley. Combinatorics and commutative algebra. Birkh¨ auser, 1996. [Stu95] B. Sturmfels. On Vector Partition Functions. Journal of Combinatorial Theory, Series A, 72:302–309, 1995. [TH86] N. V. Trung and L. T. Hoa. Affine semigroups and Cohen-Macaulay rings generated by monomials. Trans. AMS, 298(1):145–167, 1986. [Van92] M. Van den Bergh. Cohen-macaulayness of semi-invariants for tori. Trans. Am. Math. Soc., 336(2):557– 580, 1992. [vdB04a] M. van den Bergh. Non-commutative crepant resolutions. In The legacy of Niels Henrik Abel. Papers from the Abel Bicentennial Conference held at the University of Oslo, Oslo, June 3–8, 2002. Springer, 2004. [vdB04b] M. van den Bergh. Three-dimensional flops and noncommutative rings. Duke Math. J., 122(3):423–455, 2004. [Ver77] J.-L. Verdier. Cat´egories d´eriv´ees: ´etat 0 (r´edig´e en 1963, publi´e en 1976). In S´eminaire de G´eometrie ´ Alg´ebrique 4 1/2, Cohomologie Etale, volume 569 of Lect. Notes Math., pages 262–311. Springer, 1977. [Ver96] J.-L. Verdier. Des cat´egories d´eriv´ees des cat´egories ab´eliennes, volume 239 of Ast´erisque. Soc. Math. France Inst. Henri Poincar´e, 1996. [Yos90] Y. Yoshino. Cohen-Macaulay modules over Cohen-Macaulay rings, volume 146 of London Mathematical Society lecture note series. Cambridge University Press, 1990.