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I thank my parents, Kristine and Mark Parker, my brother Jeff, and my sister Dianna as well as ... The University of Texas at Austin, 2005. Supervisor: Seán Keel.
Copyright by Adam Edgar Parker 2005

The Dissertation Committee for Adam Edgar Parker Certifies that this is the approved version of the following dissertation:

AN ELEMENTARY CONSTRUCTION OF M 0,0(Pr , d)

Committee:

Se´an Keel, Supervisor

Dan Freed

Tamas Hausel

Fernando Rodriguez-Villegas

Ana-Maria Castravet

AN ELEMENTARY CONSTRUCTION OF M 0,0(Pr , d)

by ADAM EDGAR PARKER, B.S., B.A.

DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN May 2005

Dedicated to Bernadette.

Acknowledgments

I would not have been able to complete this Thesis without the help and ideas of Prof. Se´an Keel. His encouragement, guidance, and patience were crucial. In addition, he helped me become a truly mediocre poker player. I would like to thank my committee of Drs. Freed, Hausel, RodriguezVillegas, and Castravet for seeing over the defense process. I need to thank my friends Alan, Chris, Christine, Cecilia, Clay, Eduardo, Jenn, Judy, Mike, Kate, Kelly, and Kurt. Each one of you played a very important role in helping to keep me sane. I’d like to thank Dr. Hamrick, Susan Brown, Annette Hairston, Michele Keeler, and Joe G. for making my experience as Ombudsman for the math department so rewarding. In addition, Dr. Palka, Jan Baker, Nita Goldrick, Suzanne Williams, and Nancy Lamm were invaluable answering endless questions and making sure that I didn’t get lost. I thank my parents, Kristine and Mark Parker, my brother Jeff, and my sister Dianna as well as the rest of my family for all of their encouragement and support. Finally, I’d like to thank Bernadette for her patience and understanding. Thank you all.

v

AN ELEMENTARY CONSTRUCTION OF M 0,0(Pr , d)

Publication No. Adam Edgar Parker, Ph.D. The University of Texas at Austin, 2005

Supervisor: Se´an Keel

This Thesis provides an elementary construction of the Moduli Space of Stable Maps M 0,0 (Pr , d). It is constructed as a sequence of “weighted blowups along regular embeddings” of a projective variety. This is a corollary to a more general GIT construction that places M 0,n (Pr , d), P1 [n], and M 0,n into a single context.

vi

Table of Contents

Acknowledgments

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Abstract

vi

List of Tables

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List of Figures

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Chapter 1. Introduction 1.1 A Mathematical Introduction . . . . . . . . . . . . . . . . . . . . 1.2 For the Non-mathematician . . . . . . . . . . . . . . . . . . . . .

1 1 6

Chapter 2. A Quotient Approach to M 0,n (Pr , d) 2.1 The Moduli Space of Stable Maps . . . . . . . . . . 2.1.1 Basic Properties . . . . . . . . . . . . . . . . 2.1.2 The Boundary . . . . . . . . . . . . . . . . . 2.1.3 Examples . . . . . . . . . . . . . . . . . . . . 2.2 Compactifications of Degree d Maps From P1 → Pr 2.2.1 Two Different Compactifications . . . . . . . 2.2.2 Two Important Maps . . . . . . . . . . . . . 2.3 The Quotient Construction . . . . . . . . . . . . . . 2.3.1 Group Actions . . . . . . . . . . . . . . . . .

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11 11 11 19 21 24 24 27 30 30

2.3.2 Chamber Decomposition of PicG ((P1 )n × Prd ) . . . . . . . . 2.3.3 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . .

33 46

Chapter 3.

The Betti Numbers of ((P1 )n × Prd )ss /G

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Chapter 4.

Intermediate Moduli Spaces r

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4.1 Rigid Versions of M 0,0 (P × P , (d, 1)) and

Prd

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4.1.1 Constructing M 0,0 (Pr × P1 , (d, 1)) 4.1.2 Constructing Prd . . . . . . . . . . 4.2 Relating the Rigid Covers . . . . . . . . 4.2.1 The Group Action . . . . . . . . . 4.2.2 Taking the Quotient . . . . . . . .

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4.3 Factoring M 0,0 (Pr , d) . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3.1 Factoring P1 [(r + 1)d] → (P1 )(r+1)d . . . . . . . . . . . . . . 4.3.2 Factoring ϕ(t¯) : G(t¯) → Prd (t¯) . . . . . . . . . . . . . . . . .

81 83

4.3.3 Factoring ϕ¯ : M 0,0 (Pr , d) → Prd /G . . . . . . . . . . . . . . .

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Chapter 5. A Word About Stacks 5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Group Actions on Stacks . . . . . . . . . . . . . . . . . . . . . . .

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Bibliography

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Vita

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viii

List of Tables

3.1 3.2

Betti Numbers of (Pr1 )ss /G . . . . . . . . . . . . . . . . . . . . r

Comparison of Betti Numbers of M 0,0 (P , 3) and

ix

(Pr3 )ss /G

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61 61

List of Figures

2.1 2.2 2.3 2.4 2.5 2.6

A Stable Map . . . . . . . . . . . . . . . . . . . . . . . . . . A Boundary Divisor . . . . . . . . . . . . . . . . . . . . . . Self-Intersection of Divisors . . . . . . . . . . . . . . . . . . Full Dimension Boundary Component in M 1,1 (P2 , 3) . . . . . A Comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Components in M 0,1 (Pr × P1 , (d, 1)) Mapping to M0,1 (Pr , d)

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13 15 21 22 26 34

2.7

Stable Locus of M 0,3 (Pr × P1 , (2, 1)) . . . . . . . . . . . . . . .

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4.1

Relating GIT Quotients . . C¯ ⊂ C . . . . . . . . . . . . The GIT Quotients . . . . . The First Blow Down . . . . Factoring ϕ(t¯) . . . . . . . . Factoring ϕ . . . . . . . . . Factoring ϕ¯ . . . . . . . . .

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4.2 4.3 4.4 4.5 4.6 4.7

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Chapter 1 Introduction 1.1

A Mathematical Introduction

Given a projective space Pr and a class d ∈ A1 (Pr ) ∼ = Z, a n-pointed, stable map of degree d consists of the data (µ : C → Pr , {pi}ni=1 ) where: • C is a complex, projective, connected, reduced, n-pointed, genus 0 curve with at worst nodal singularities. • {pi } are smooth points of C. • µ : C → Pr is a morphism. • µ∗ [C] = dl, where l is a line generator of A1 (Pr ). • If µ collapses a component E of C to a point, then E must contain at least three special points (nodes or marked points). We say that two stable maps are isomorphic if there is an isomorphism of the domain marked curves f : C → C ′ that commutes with the morphisms to Pr . Then there is a projective coarse moduli space M 0,n (Pr , d) that parametrizes stable maps up to isomorphism [5].

1

The problem discussed in this Thesis involves finding an elementary construction of M 0,0 (Pr , d). Specifically, we would like to construct M 0,0 (Pr , d) as sequence of blow-ups of some projective variety. One benefit of such a construction is the ability to compute the Chow ring of M 0,0 (Pr , d), as Keel’s Theorem 1 from the appendix of [16] gives the Chow ring of a blow up. Such a project was carried out for M 0,n in [16], and for X[n] in [7]. These are interesting models for us, as M 0,n (Pr , 0) = M 0,n × Pr and M 0,n (P1 , 1) = P1 [n]. There are two issues that make such a construction problematic, in general, for stable maps. First, M 0,0 (Pr , d) is not smooth. It has singularities at points corresponding to maps with nontrivial automorphisms. These singularities are acceptably mild, as the stability condition on our morphisms forces their automorphism groups to be finite. Thus, M 0,0 (Pr , d) has finite quotient singularities and is topologically an orbifold. However, M 0,0 (Pr , d) is actually smooth when considered as a stack, and so at best we may hope for a stack analogue of a sequence of blow ups mentioned above. If such a construction exists, care must be taken in constructing the stack analogue of Keel’s Theorem to get the Chow ring. The second issue seems more serious. There are no known maps from M 0,0 (Pr , d) to anything nice, and a birational map from M 0,0 (Pr , d) is exactly what is needed to carry out the above project. In the case of M 0,n , there is the natural map M 0,n → M 0,n−1 × P1 (forgetting the nth marked point and collapsing unstable components) which Keel factors [16]. In [7], X[n] is constructed as a sequence of blow ups of X n , and so there is an obvious blow down X[n] → X n .

2

When there are marked points on the domain curve, the corresponding moduli space M 0,n (Pr , d) does have maps to other varieties. There are forgetful morphisms M 0,n (Pr , d) → M 0,n−1 (Pr , d) completely analogous to the curve case above. There are also evaluation morphisms evi : M 0,n (Pr , d) → Pr obtained by looking at the image of the ith marked point. The product of evaluation morphisms M 0,n (P1 , 1) → (P1 )n is exactly the map from [7]. There is, however, one case of a moduli space without marked points having a map from it. Consider the graph space M 0,0 (Pr × P1 , (d, 1)) that is defined completely analogously to M 0,0 (Pr , d) above. In [9], Givental defines a projective morphism ϕ : M 0,0 (Pr × P1 , (d, 1)) → Prd := P((H 0 (P1 , O(d))r+1 ). Set theoretically, consider a point in M 0,0 (Pr × P1 , (d, 1)). There is a representative [µ : C → Pr × P1 of bi-degree (d, 1)] and a component C0 ⊂ C such that µ|C0 is the graph of r + 1 degree d′ homogenous polynomials (f0 , . . . , fr ) with no common zero. On the other components C1 , . . . , Cs , µ has degree (di , 0) respectively, and d1 + · · · + ds = d − d′ . Thus µ sends Ci into Pr × zi ⊂ Pr × P1 . Let h be a degree d − d′ form that vanishes at each zi with multiplicity di. Then by reading off the coefficients, we get a point in projective space. ϕ(µ) = [f0 · h, f1 · h, . . . , fr · h] ∈ Prd . We extend this to a birational projective morphism φ : M 0,n (Pr × P1 , (d, 1)) → (P1 )n × Prd . 3

by using the i-th evaluation morphism ev

π

2 M 0,n (Pr × P1 , (d, 1)) −−−i→ Pr × P1 −−− → P1

as the map to the i-th P1 . Both M 0,n (Pr ×P1 , (d, 1)) and (P1 )n ×Prd can be thought of as compactifications of the space of n-pointed, degree-d parametrized maps from P1 → Pr . Both also carry natural G = SL2 (C) actions. It turns out that φ is equivariant with respect to these actions. And while M 0,0 (Pr , d) may not have any maps from it, there are maps to M 0,0 (Pr , d). The most important for our purposes is the surjective forgetful map f : M 0,0 (Pr × P1 , (d, 1)) → M 0,0 (Pr , d) that forgets the map to P1 and collapses unstable components. There is the analogous forgetful map for the pointed moduli spaces M 0,n (Pr × P1 , (d, 1)) → M 0,n (Pr , d). As the action of G on M 0,n (Pr , d) is trivial, f is G invariant and so must be some type of quotient of M 0,0 (Pr × P1 , (d, 1)). It is, in fact, a GIT quotient as we see from the following general Theorem. Theorem 1.1.1. Let E be an effective divisor such that −E is φ-ample. Take a linearized line bundle L ∈ PicG ((P1 )n × Prd ) such that ((P1 )n × Prd )ss (L) = ((P1 )n × Prd )s (L) 6= ∅. Then for each sufficiently small ǫ > 0, the line bundle L′ = φ∗ (L)(−ǫE) is ample and (M 0,n (Pr × P1 , (d, 1)))ss(L′ ) = (M 0,n (Pr × P1 , (d, 1)))s(L′ ) = φ−1 {((P1 )n × Prd )ss (L)}. 4

There is a canonical identification (M 0,n (Pr × P1 , (d, 1)))s (L′ )/G = M 0,n (Pr , d) and a commutative diagram f

(M 0,n (Pr × P1 , (d, 1)))s(L′ ) −−−→   φy ((P1 )n × Prd )s (L)

M 0,n (Pr , d)   y

−−−→ ((P1 )n × Prd )s (L)/G

The proof proceeds as follows. First, given a G-linearlized ample line bundle on (P1 )n × Prd , the semi-stable and stable loci are found for the G action on Prd . For certain line bundles, the semi-stable locus and stable locus will agree. A Theorem of Hu ([13]) then gives the first part of this Theorem. Showing the second equality involves showing that both sides have the same Picard number. Finally, since any other G invariant morphism (such as M 0,n (Pr × P1 , (d, 1)) → ((P1 )n × Prd )/G) will factor through the GIT quotient, we get the commutative diagram. Notice that as special cases of this Theorem, we get GIT constructions of M 0,n × Pr (when d = 0), the Fulton-MacPherson space P1 [n] (when d = 1), and the Grassmannian of lines G(1, r) (when d = 1 and n = 0). Finally, in the case when n = 0 we get what we wanted: a map from M 0,0 (Pr , d) to a projective variety. In [25], Andrei and Magdalena Mustat¸ˇa carry out the above project (computing Chow rings) for M 0,1 (Pr , d). The morphism that they consider comes from the evaluation morphism ev mentioned above. However, they place it into a different setting. They embed M 0,1 (Pr , d) into M 0,0 (Pr × P1 , (d, 1)) as 5

a divisor by attaching a parametrized P1 to the domain curve at the marked point. Then the image under ϕ will be exactly a Pr ֒→ Prd , as all of the interesting part of the map lies off the parametrized component. The factorization into blow-ups is then modeled from the sequence of blow ups of P1 [(r + 1)d] described in [7]. As mentioned above, however, these are not true blow ups factoring ϕ, but rather stack analogues called “weighted blow ups of a regular local embedding”. Instead of restricting the sequence of blow ups to factor M 0,1 (Pr , d) → Pr , it is more natural to look at the complete factorization of ϕ : M 0,0 (Pr × P1 , (d, 1)) → Prd that is a corollary of their construction. By applying arguments similar to those in the proof of Theorem 1.1.1, we have: Theorem 1.1.2. There is a sequence of intermediate moduli spaces that factor ϕ¯ : M 0,0 (Pr , d) → (Prd )ss /G such that the map between successive intermediate spaces is a weighted blow up of a regular local embedding.

1.2

For the Non-mathematician

What follows is meant as an introduction in the most general of terms. No details are given, and much of what is said is incorrect in the form stated. The hope is to give a flavor of what this Thesis attempts to accomplish without any prerequisite knowledge. In mathematics, it is common to take some type of objects and ask, “What can you say about these?” This can be a difficult problem because 6

without a way of relating the objects to each other, we’re left trying to describe a (possibly huge) set of objects, and finding structure here can be hard. For example, say the “geometric” objects are colors. We can say things about colors because we can place them into a context that relates them to each other (like a rainbow, or a color wheel) such that similar colors are “close” to each other. We can test that “maize” + “sky blue” = “grass green.” Then, because we have a context relating colors to each other, we can say “in general” that “yellows” + “blues” = “greens” without testing every single shade of yellow, and blue. If we didn’t have the rainbow or color wheel, we couldn’t make that statement. We could only find relations that we could test individually. Thus, if I am given a type of object to study, I want to find an object X such that every point on X corresponds to one of the original objects I was given to study. I want X to be a geometric object in its own right, and I want similar geometric objects to be represented by points on X that are close to each other. Then X is a moduli space for our objects. Think of X as a menu at a restaurant. The entries on the menu correspond to foods (but the entries aren’t foods themselves), and similar foods are placed close to each other (appetizers, salads, desserts). Mathematically, moduli spaces are everywhere and very useful. In this Thesis, the objects are degree d functions [µ : C → Pr ], where C is a curve and Pr is a well known geometric object (though it’s not important what it is). By “degree d” I mean that these maps look like degree d polynomials. The curve C need not be smooth, but the singularities that it has can’t be too wild. For example, we allow the curve to cross itself (forming an “X”)

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but do not allow the curve to pinch ( forming a “V”). In 1994, Maxim Kontsevich introduced the moduli space M 0,0 (Pr , d) that parametrized degree d stable maps up to the equivalence [16]. The construction is carried out in detail in [5]. Though we have a construction of this space we don’t know much about it. There is currently a lot of work being done to change this. A typical method of learning about any geometric object (such as M 0,0 (Pr , d)) is to find a different geometric object Y , and a map M 0,0 (Pr , d) → Y such that Y is similar, but “smaller.” By this I mean that the two spaces look identical almost everywhere, except there is some sub set of M 0,0 (Pr , d) that is collapsed in Y . Perhaps there is a line in M 0,0 (Pr , d) that is squashed to a point in Y . Such a map is called a regular, bi-rational map. Once we have such a regular, bi-rational map, we examine it closely and hope that we can factor it into intermediate steps in such a way that the intrinsic properties of Y can be kept track of at each step. Thus, if we know the properties of Y and we know how they change at each step, then we can determine the properties of M 0,0 (Pr , d). The problem is that in the case of M 0,0 (Pr , d), there wasn’t a candidate for the space Y . Even worse, there are essentially NO regular, bi-rational maps from M 0,0 (Pr , d) to even consider. This Thesis starts by finding one. Instead of considering M 0,0 (Pr , d) directly, we examine a different geometric object M 0,0 (Pr × P1 , (d, 1)) (it isn’t important what it is). There is a map f from 8

M 0,0 (Pr × P1 , (d, 1)) to M 0,0 (Pr , d), and luckily, there is also a regular, birational map ϕ : M 0,0 (Pr × P1 , (d, 1)) → Prd where Prd is a very easy space to understand. We can sum this up with the diagram: f

M 0,0 (Pr × P1 , (d, 1)) −−−→ M 0,0 (Pr , d)   ϕy Prd

And, it turns out that exactly what you hope would happen, actually happens. There is a space Y that goes in the lower right of the above square forming a square diagram. Thus we have f

M 0,0 (Pr × P1 , (d, 1)) −−−→ M 0,0 (Pr , d)     ϕy ϕ ¯y Prd

−−−→

Y

such that ϕ¯ is a regular, bi-rational map like ϕ is. This puts us in the situation mentioned above. The second part of my research involves examining the map ϕ¯ carefully and factoring it into intermediate steps. In [25], the map ϕ on the left of the above diagram is (essentially) factored into a sequence of intermediate steps such that the map between two successive spaces is a “weighted blow up of a local embedding.” In a similar construction to what is done above, I am able use them to factor ϕ, ¯ as a sequence of steps. This allows us to do several things. First of all, it reduces some questions about M 0,0 (Pr , d) to questions about M 0,0 (Pr × P1 , (d, 1)) which in some sense is an easier space to understand. Secondly, there are some classical geometric spaces for which similar square diagrams above are known. Choosing r

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and d carefully gives those constructions as special cases of what I’ve proven. Finally, it lays the groundwork for the program outlined above.

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Chapter 2 A Quotient Approach to M 0,n(Pr , d) 2.1

The Moduli Space of Stable Maps

This section surveys basic results and definitions concerning the moduli space of stable maps that will be used throughout this Thesis. Some examples are included for intuition. We also standardize notation. All work will be done over C. The ideas come primarily from [5]. 2.1.1

Basic Properties

As with any moduli problem, we need to discuss two things before defining the moduli space of stable maps. First, we give a precise definition of what objects we would like to study (stable maps), and a notion of what it means put these objects into a family. Secondly, we will give an equivalence relation on the set of all families over a fixed base ([11]). The moduli space then classifies the families of our objects up to this equivalence relation, in a natural way. We begin with the following definition. Definition 2.1.1. ([5]). Given algebraic schemes S and convex X over C, nonnegative integers g, n, and a class β ∈ A1 (X), define a stable map over S

11

from n-pointed, genus g curves to X, representing β, to be the following data (π : C → S, {pi }1≤i≤n , µ : C → X) where: 1. π : C → S is a flat, projective map. 2. {pi } are sections of π. 3. Each geometric fiber (Cs , p1 (s), . . . , pn (s)) is a projective, connected, reduced n pointed curve of genus g with at worst nodal singularities. 4. µ : C → X is a morphism. 5. µ∗ [C] = β. 6. If µ collapses a component E of genus 0 of Cs to a point, then E must contain at least three special points (either marked points or nodes.) 7. If µ collapses a component F of genus 1 of Cs to a point, then F must contain at least one special point. Conditions 6 and 7 above will sometimes be referred to as the stability conditions. Notice that the domain of a stable map need not be a stable curve in the sense of Deligne-Mumford. (See [5], or [16] for an introduction.) Now that we have the objects that we wish to study, we need our equivalence relation. Two families of maps are equivalent if they are isomorphic. Definition 2.1.2. An isomorphism between the maps (π : C → S, {pi }, µ : C → X) and (π ′ : C ′ → S, {p′i }, µ′ : C ′ → X) is an isomorphism of domain 12

X Cs

S

Figure 2.1: A Stable Map curves f : C → C ′ such that the marked points are mapped to the marked points, µ′ ◦ f = µ, and π ′ ◦ f = π. C

f

′ ................................................................................................................................... .... . ... .... ....... . . . . ....... . . ... ... ... .... ... .... ′ .......... ...... ... .... .. ... .... . .. . . . .... ... .. ... ... ... ... .... .... ... ... ... ... .. . ... . . . . . . .... ... .. .. ... ... ... .. ..... ...... ... ... ...... ....... ... ... .. ... . ... ... ... ... ′ ... .. ... .. . ... . ... ... .. ... . . ... ... ... ... ... ... .. .. ... . . ... .. ... .. . .......... .......... ..

C

µ

π

µ

X

π

S Then M g,n (X, β) is the set of all isomorphism classes of n-pointed, genus-g stable maps with target X, representing β. M g,n (X, β) has more structure than just a set, as we see from the following Theorem. Theorem 2.1.1. [5] The space M g,n (X, β) is a projective coarse moduli space for the moduli functor Mg,n (X, β) : {Schemes} → {Sets} 13

given by: Mg,n (X, β)(S) = {The set of all families of stable maps from n-pointed, genus-g curves to X, representing β, up to S-isomorphisms.} This means that there is a projective scheme M g,n (X, β) together with a natural transformation of functors η : Mg,n (X, β) → HomSch (∗, M g,n (X, β)) satisfying properties 1. η(Spec(C)) : Mg,n (X, β)(Spec(C)) → Hom(Spec(C), M g,n (X, β)) is a set bijection. 2. If Z is a scheme and ψ : Mg,n (X, β) → Hom(∗, Z) is a natural transformation of functors, then there exists a unique morphism of schemes γ : M g,n (X, β) → Z such that ψ = γ˜ ◦ η, where γ˜ : Hom(∗, M g,n (X, β)) → Hom(∗, Z) is the natural transformation induced by γ. In other words, given any stable map (π : C → S, {pi }, µ : C → X), there is a unique morphism S → M g,n (X, β), taking s ∈ S to the isomorphism class of the fiber over s. This is distinguished from a fine moduli space as there is no mention of a universal family over M g,n (X, β) (compare with [11]). Remark 2.1.1. In general, the moduli space of stable maps is not well behaved ([26] or Example 2.1.2), and so we won’t study the full generality defined above. 14

Instead we restrict to the cases g = 0 and X = Pr or Pr × P1 . In the case when the range is a projective space, then β ∈ A1 (Pr ) = Z will be called the degree of µ and written d. When the range is Pr × P1 , then β ∈ Z ⊕ Z = A1 (Pr × P1 ) is the bi-degree of µ and written (d, d′ ). We collect some basic results about M 0,n (Pr , d) here. 1. The locus of maps having an irreduciable domain will be denoted by Mg,n (X, β) ⊂ M g,n (X, β). The complement of this is called the boundary, and will be discussed in more detail in the next subsection.

3

4

2

5 1

d2 d 1

Figure 2.2: A Boundary Divisor

2. From Theorems 1 and 2 in [5], we know that M 0,n (Pr , d) is a normal, projective variety of dimension (r + 1)(d + 1) − 4 + n. In general, when X is projective, nonsingular, and convex dim(M 0,n (X, β)) = dim(X) +

Z

c1 (TX ) + n − 3.

β

3. In addition, we know that M 0,n (Pr , d) is locally the quotient of a smooth variety by a finite group. Indeed, in the construction of M 0,n (Pr , d) ([5]), 15

Fulton and Pandharipande make explicit both the smooth variety and group. Given a basis t¯ = (t0 , . . . , tr ) of H 0 (Pr , OPr ), they define a t¯-ridgid stable family of degree d maps from n-pointed, genus 0 curves to Pr as the following data (π : C → S, {pi }1≤i≤n , {qi,j }0≤i≤r,1≤j≤d, µ : C → Pr ) where (a) (π : C → S, {pi }, µ) is an n-pointed, genus-0 stable map as in Definition 2.1.1. (b) (π : C → S, {pi }, {qi,j }) is a flat, projective faimly of n + (r + 1)d pointed stable curves in the sense of Deligne-Mumford. (c) For 0 ≤ i ≤ r there is an equality of Cartier divisors µ∗ (ti ) = qi,1 + qi,2 + · · · + qi,d The obvious moduli functor of t¯-ridgid stable maps is represented by a smooth algebraic variety M (t¯) := M 0,n (Pr , d, ¯t). The finite group G = (Sd )r+1 acts on the sections {qi,j } and thus on M (t¯). The quasi-projective quotient M (t¯)/G = Ut¯ ⊂ M 0,n (Pr , d) is an open set in the moduli space of stable maps. For different values of t¯, the open sets Ut¯ cover and patch together to give us M 0,n (Pr , d). This construction will be mimicked in Chapter 3. 4. In fact, [M 0,n (Pr , d)] is a smooth Deligne-Mumford Stack (geometrically an orbifold) that represents the stacky functor (when we refer to the 16

stack, we put braces around our schemes). [M0,n (Pr , d)] : {Schemes} → {Groupoids} Specifically, [M 0,n (Pr , d)] is a category, with objects and morphisms Ob([M 0,n (Pr , d)]) = { All families of stable maps from n-pointed, genus-0 curves to Pr , of degree d. }

Mor([M 0,n (Pr , d)]) : (π : C → S, µ : C → Pr {pi }) → (π ′ : C ′ → S ′ , µ′ : C ′ → Pr {p′i }) is a pair of maps α : C ′ → C and β : S ′ → S making the following diagram commute

α

C ′ −−−→ C     ′ π πy y β

S ′ −−−→ S. Moreover, we require that C ′ is isomorphic to the pullback of C by β and µ = µ′ ◦ α. There is a functor [M 0,n (Pr , d)] → {Schemes} that is just projection to the base. If we restrict to families over a fixed base, (i.e. look at the fiber of the above functor over a fixed scheme), the corresponding restriction is a groupoid, or a category where all morphisms are isomorphisms. See Chapter 4 for the definition and additional citations on stacks. We won’t take this approach in this chapter. 17

5. In [18] and [30], it is shown that M 0,n (Pr , d) is connected. Moreover, because all singularities in M 0,n (Pr , d) are finite quotient in type, we see that connectedness is equivalent to irreducibility. 6. We also have that M 0,n (Pr , d) is Q-factorial (which means that a power of every Weil divisor is Cartier). This will be important later, and follows from the construction in 3 above, along with the following general proposition. Proposition 2.1.2. Suppose Y is locally the quotient of a smooth variety by a finite group G. Then Y is Q-factorial. Proof. We assume that Y is the quotient of a smooth variety X by a finite group G. Let π : X → Y be the quotient. Let D be a (prime) divisor on Y , ¯ be the Weil divisor associated to the scheme π −1 (D). Since X is and let D ¯ is a line bundle. smooth, we know every Weil divisor is Cartier, and so OX (D) I will use the norm to push forward the transition functions of this line bundle. Recall the norm N : K(X) → K(X)G Y Y N(f (x)) = σ(f (x)) = f (σ −1 (x)). σ∈G

In addition, we have that N :

π∗ (O∗X )

σ∈G



O∗Y

. For assume since π is affine that

X = Spec A, and Y = Spec AG are affine. Then, if f ∈ A is a nowhere zero function on X, N(f ) is the product of regular functions that are nowhere zero. As long as AG is a domain, then N(f ) ∈ O∗Y . Note that we can think of K(X)G as sitting inside of K(X) via π ∗ . Then for f ∈ K(X)G ⊂ K(X) , N(f ) = f |G| . 18

Choose an open cover {Uα } of Y such that {π −1 (Uα )} is an open cover ¯ Such an open cover exists by Lemma B in [23], of X that trivializes OX (D). (pg 72). If fα,β ∈ O∗ (π −1 (Uα ∩ Uβ )) are the transition functions, then we ¯ to be the line bundle on Y given by the open cover Uα and define N(O(D)) the transition functions N(fα,β ). Now, take an open subset V ⊂ Y such that Y \ V has codimension ¯ V = ≥ 2, and that D|V is Cartier. By the note above, we have that N(O(D))| OY (|G|D)|V . Since they agree on an open set, the agree everywhere, and ¯ ∈ Pic(Y ), which means that |G|D is Cartier. OY (|G|D) = N(O(D))

2.1.2

The Boundary

We describe the boundary components of M 0,n (Pr , d) using the notation of [5]. Let N1 ∪ N2 be a partition of {1, 2, . . . , n} and d1 + d2 = d. We define D(N1 , N2 , d1 , d2 ) to be the fiber product M 0,N1 ∪{∗} (Pr , d1 ) ×Pr M 0,N2 ∪{∗} (Pr , d2 ) −−−→ M 0,N2 ∪{∗} (Pr , d2 )    ev y y M 0,N1 ∪{∗} (Pr , d1 )

−−−→ ev

Pr

where ev is evaluation at the special marked point (∗). To formally define ev, we define a natural transformation θ : M0,n∪{p} (Pr , d) → Hom(∗, Pr ). by: given a scheme S, we let F = (π : C → S, {p}, µ : C → Pr ) be an element of the set M0,n∪{p} (Pr , d)(S). Then we define an element of Hom(S, Pr ) by θ(S)(F ) = µ ◦ p ∈ Hom(S, Pr ) 19

By the universal properties of a coarse moduli space (Definition 2.1.1), we get a unique morphism ev : M 0,n∪{p} (Pr , d) → Pr . The above fiber product then says that the maps take the same value in Pr on the extra marked point under the maps ev. The basic Theorem is the following. Theorem 2.1.3. [5] The boundary of M 0,n (Pr , d) is a divisor with normal crossings (up to a finite group quotient). Again, we collect some useful results. 1. Note that D(N1 , N2 , d1 , d2 ) = D(N2 , N1 , d2, d1 ). 2. It’s clear from the stability condition in Definition 2.1.1 that M 0,1 (Pr , 0) is empty. As such, we don’t have a boundary when d ≤ 1. In fact, it’s not hard to see that M 0,0 (Pr , 1) = M0,0 (Pr , 1) is smooth. 3. The boundary divisors self-intersect for degree large enough. For example, look at the boundary divisor D(2, 3) ⊂ M 0,0 (Pr , 5). . The divisor D(2, 3) self intersects in the codimension-2 locus of curves with three components where the ending curves have degree 2, and the middle component has degree 1. See Figure 3. 4. In [30], it is shown that the divisors D(N1 , N2 , d1 , d2 ) ∈ M 0,n (Pr , d) are irreduciable. 5. In the case of the graph space M 0,n (Pr ×P1 , (d, 1)) (which we will be concerned with soon) we have boundary divisors D(N1 , N2 , (d1, 1), (d2 , 0)) ∈ 20

2

1

2

Figure 2.3: Self-Intersection of Divisors M 0,n (Pr ×P1 , (d, 1)). These enjoy all of the same properties as the boundary in M 0,n (Pr , d). When no confusion arises, we will write D(N1 , N2 , d1 , d2 ) for D(N1 , N2 , (d1, 1), (d2 , 0)). Note that with this notation that D(N1 , N2 , d1 , d2 ) 6= D(N2 , N1 , d2 , d1 ). 2.1.3

Examples

Example 2.1.2. The moduli space of stable maps can be badly behaved. Even when X = Pr , the open locus Mg,n (Pr , d) ⊂ M g,n (Pr , d) may not be dense. For example, look at M 1,1 (P2 , 3). By [5] we see that this space has dimension 11. However, its boundary contains a component f : C → P2 , where the domain curve is reducible C = C0 ∪ C1 with C0 of genus 0 with one marked point, and C1 of genus 1. The map is of degree 3 on C0 and of degree 0 on C1 . In other words, the boundary contains a copy of M 0,2 (P2 , 3) × M 1,1 ⊂ ∂M 1,1 (P2 , 3)

21

which has dimension 11 as well. This is why restrict ourselves to g = 0. See [26] for more pathologies. d2 = 3 d1 = 0

g=0

g=1 P2

Figure 2.4: Full Dimension Boundary Component in M 1,1 (P2 , 3) Example 2.1.3. Recall the familiar case of elliptic curves M 1,1 [11]. Here, the j-line is not a fine moduli space because of the existence of elliptic curves with automorphisms beyond inversion. Similarly, M 0,n (Pr , d) fails to be a fine moduli space because of the existence of maps with nontrivial automorphisms. In the n = 0 case, consider the map f : P1 → P1 f [x0 : x1 ] = [x20 : x21 ]. Then the map sending [x0 : x1 ] → [−x0 , x1 ] is a non trivial automorphism of order 2 of this map. Moreover, notice that this also shows that M 0,0 (Pr × P1 , (d, 1)) isn’t smooth for d ≥ 2. For, take a curve C = C0 ∪ C1 with two components. Take the degree (d, 1) map to P1 × P1 that is (d − 2, id) on C0 , and (f, 0) on C1 (f as above). Then have the above automorphism act just on C1 . M 0,0 (Pr × P1 , (d, 1)) has pushed the singularities of M 0,0 (Pr , d) into the boundary. 22

Example 2.1.4. In the case when d = 0 the stability condition in Definition 2.1.1 is equivalent to the Deligne-Mumford stability condition on M 0,n and we have M 0,n (Pr , 0) = M 0,n × Pr . Example 2.1.5. In the case when d = 1, n = 0 we have that M 0,0 (Pr , 1) = G(1, r) is the Grassmannian of lines in Pr . Both of these spaces are moduli spaces for lines in Pr . The universal property of M 0,0 (Pr , 1) gives a map G(1, r) → M 0,0 (Pr , 1), while universal properties of the Grassmannian gives a map in the other direction. Both spaces are smooth, hence isomorphic. Example 2.1.6. In the case when d = 1 and r = 1, we have that M 0,n (P1 , 1) = P1 [n] where P1 [n] is the Fulton-MacPherson compactification of n points on P1 . By [7], there is a universal family p : P[n + 1] → P[n]. whose fiber is is a tree of genus 0, n-pointed curves. One of the components is parametrized, and all of the additional components must have at least three special markings on it (marked points or nodes) giving trivial automorphism of those components. This is M 0,n (P1 , 1). Similarly, we see that M 0,n (Pr × P1 , (0, 1)) = Pr × P1 [n].

23

2.2

Compactifications of Degree d Maps From P1 → Pr

Suppose that we wanted to compactify the space of n-pointed degree-d morphisms from P1 → Pr . Perhaps after the above discussion, one would expect that M 0,n (Pr , d) correctly compactifies these objects. However, since we quotient out by isomorphisms, we only get degree d un-parametrized pointed morphisms to Pr . Here we discuss two spaces that do correctly answer this question. 2.2.1

Two Different Compactifications

On one hand, an n-pointed, degree-d morphism f is given by (r + 1) homogenous degree d polynomials in two variables, along with a choice of n points on the domain P1 . In the notation of [27], these maps correspond to the basepoint free locus ((P1 )n \ ∆) × P(U(1, r, d)) ⊂ (P1 )n × P(

r M

H 0 (P1 , OP1 (d)))

0

= (P1 )n × P(

r M

Symd (C2 ))

0

1 n

:= (P ) ×

Prd .

Once we pick coordinates on P1 , we can consider a closed point on the basepoint free locus as [x1 : y1 ] × . . . , [xn : yn ] × [f0 (x, y) : f1 (x, y) : · · · : fr (x, y)] where [x1 : y1 ] 6= [xn : yn ], the fj don’t have any common roots, and scaling doesn’t change the map. The coefficients of these fj determine a point in proj jective space Prd ∼ = P(r+1)(d+1)−1 . We will sometimes write ai for the coefficient

24

xd−i y i on fj (after choosing the obvious coordinates on Prd .) We thus have a simple compactification by allowing the r + 1 forms to have common roots, and allowing the n points to come together. This space is sometimes referred to as the linear sigma model. There is another, less simple (non-linear) compactification of ((P1 )n \ ∆) × P(U(1, r, d)). It is clear that this set equals M0,n (Pr × P1 , (d, 1)), and thus M 0,n (Pr × P1 , (d, 1)) provides another compacitification. Here, when we reparametrize the source, the image doesn’t change ([2]) as desired. This space will occupy much of this chapter. Definition 2.2.1. We will refer to the domain curve C for a map in M 0,n (Pr × P1 , (d, 1)) as a comb. There is an obvious distinguished component C0 on which µ|C0 will be of degree (d′ , 1). We will call this component the handle. The other components fit into teeth Ti , which are (perhaps reducible) genus-0, ni -pointed curves meeting C0 at unique points qi . See Figure 2.2.1. Moreover we can choose a representative of each equivalence class so that the degree 1 map above is the identity. Proposition 2.2.1. For any equivalence class [µ : C → Pr × P1 , {pi}] ∈ M 0,n (Pr × P1 , (d, 1)), there is a representative of this class such that µ|C0 is the graph of r + 1 degree d′ forms (f0 , . . . , fr ) with no common root. Proof. In other words, we would like to assume that the degree 1 part is the identity. Write C = C0 ∪ Ti , where the teeth Ti meet the handle C0 at the points qi . Say that µ|C0 is given by two maps (µ1 , µ2) of degree (d′ , 1). Consider a new marked curve C ′ = (µ2 ◦ C0 ) ∪ Ti ,. Now Ti is now attached to C0 at 25

7

Handle

1

6 (d , 0 ) 1 q

T1

1 2 (d 2 , 0 ) 3 5 q2

Teeth

T

2

(d 3 , 0)

4

(d’ , 1 )

q 3

T3

Figure 2.5: A Comb the point µ2 (qi ), and any marked points on C0 are moved as well. As marked curves, C and C ′ are isomorphic. We define a map [µ′ : C ′ → Pr × P1 , {µ(qi)}] which is (µ1 ◦ µ−1 2 , id) on µ2 ◦ C0 , and agrees with µ on the other components. This map is isomorphic to µ as they fit into the following diagram. C

′ ................................................................................................................................... .... ... .... ... . . ... . .. .... .... ′ .......... .... .. ... ... .... . . . .... .. .... .... ... .... .... ... .... .... . . . ...... .... ....... ......

C

µ

µ

Pr × P1

26

2.2.2

Two Important Maps

We introduce two maps that will be very important in the remainder of this Thesis. The first is essentially due to Alexander Givental and relates the non-linear and linear compactifications described above. Proposition 2.2.2. There is a projective birational morphism φ : M 0,n (Pr × P1 , (d, 1)) → (P1 )n × Prd Proof. This is a product of projective morphisms. The map to the ith P1 is the composition of the evaluation morphism with projection onto the P1 component: ev

π

2 M 0,n (Pr × P1 , (d, 1)) −−−i→ Pr × P1 −−− → P1 .

The map to Prd is the composition of a morphism that forgets the marked points, along with the so-called Givental morphism. f orget

ϕ

M 0,n (Pr × P1 , (d, 1)) −−−→ M 0,0 (Pr × P1 , (d, 1)) −−−→ Prd That the first arrow is a morphism follows from the functoral properties of M 0,n (Pr × P1 , (d, 1)). It’s projective because M 0,n (Pr × P1 , (d, 1)) is. In [9], Givental explicitly constructs the locally free sheaf on M 0,0 (Pr × P1 , (d, 1)) that gives ϕ. We describe this map set theoretically here. Consider a point in M 0,0 (Pr × P1 , (d, 1)). By Proposition 2.2.1, there is a representative (µ : C → Pr × P1 of bi-degree (d, 1))

27

and a component C0 ⊂ C such that on C0 , µ can be represented by r +1 degree d′ polynomials (f0 , . . . , fr ) with no common zero. On the other components C1 , . . . , Cs , µ has degree (di , 0) respectively, and d1 + · · · + ds = d − d′ . Thus µ sends Ci into Pr × zi ⊂ Pr × P1 . Then consider a degree d − d′ polynomial g that vanishes at each zi with multiplicity di . Then we define ϕ(µ) = [f0 · g, f1 · g, . . . , fr · g] ∈ Prd where we read off the coefficients to obtain the point in projective space. Because the product of projective morphisms is projective, we are done. The second map that we will be interested in is the “forgetful” morphism f : M 0,n (Pr × P1 , (d, 1)) → M 0,n (Pr , d) defined by forgeting the map to P1 and collapsing any components that become unstable. Do not confuse this map with the above map M 0,n (Pr × P1 , (d, 1)) → M 0,0 (Pr × P1 , (d, 1)) that forgets the marked points. The following general proposition shows that f is a morphism. Proposition 2.2.3. Let g : X → Y be a morphism of schemes. Then we have a morphism g∗ : M g,n (X, β) → M g,n (Y, g∗ β) Proof. This follows immediately from the universal property of coarse moduli spaces. In Theorem 2.1.1, let Z = M g,n (Y, g∗ β). Then define a natural 28

transformation of functors ψ : Mg,n (X, β) → Hom(∗, M g,n (Y, g∗ β)). as follows: given an isomorphism class of a family of stable maps over S [π : C → S, {pi }1≤i≤n , µ : C → X] ∈ Mg,n (X, β)(S) we compose with the map g (and perhaps collapse unstable components) [π : C → S, {pi }1≤i≤n , g ◦ µ : C → Y ] ∈ Mg,n (Y, g∗ β)(S). Now since M g,n (Y, g∗ β) is a coarse moduli space for the above functor, we know that there is a natural transformation of functors η : Mg,n (Y, g∗β) → HomSch (∗, M g,n (Y, g∗ β)) and so we get an element of Hom(S, M g,n (Y, g∗ β)). Then by the universal property again (this time for M g,n (X, β), we get a unique morphism of schemes g∗ : M g,n (X, β) → M g,n (Y, g∗ β).

This can be summarized in the diagram of morphisms: f

M 0,n (Pr × P1 , (d, 1)) −−−→ M 0,n (Pr , d)   φy (P1 )n × Prd

29

2.3

The Quotient Construction

In this section we construct the morphism M 0,0 (Pr , d) → Prd mentioned in the introduction. The existence of this morphism will be a corollary to a general construction of M 0,n (Pr , d) as a GIT quotient of M 0,n (Pr × P1 , (d, 1)). We do this in three parts. First we explicitly define actions of a group G on M 0,n (Pr × P1 , (d, 1)) and (P1 )n × Prd , showing that φ from Theorem 2.2.2 is equivariant with respect to these actions. Secondly we describe the chamber decomposition for the ample G- linearized line bundles PicG ((P1 )n × Prd ). For different ample line bundles, we explicitly describe the corresponding stable and semi-stable loci and describe how to pull these loci back under φ to M 0,n (Pr × P1 , (d, 1)). We prove our Theorem in the third subsection. 2.3.1

Group Actions

Here we make explicit actions of G = SL2 (C) on the two different compactifications of degree d maps to Pr from the previous section. 1. We start with G acting on the linear sigma model. The action is given by G × ((P1 )n × Prd ) → (P1 )n × Prd

g · [[x1 : y1 ] × · · · × [xn : yn ] × f0 (x : y), . . . , fr (x : y)] = [g[x1 : y1 ] × · · · × g[xn : yn ] × f0 ◦ g −1 (x : y), · · · : fr ◦ g −1 (x : y)] where g and g −1 act on [x : y] by matrix multiplication. 2. The action on M 0,n (Pr × P1 , (d, 1)) is induced by the action on the image 30

P1 . Namely we have G × M 0,n (Pr × P1 , (d, 1)) → M 0,n (Pr × P1 , (d, 1)) g · [µ1 × µ2 : C → Pr × P1 ] → [µ1 × g ◦ µ2 : C → Pr × P1 ] Proposition 2.3.1. The above action on M 0,n (Pr × P1 , (d, 1)) is well defined up to isomorphism. Proof. We need to check that for two different representations of the same equivalence class, then g sends them to the same equivalence class in M 0,0 (Pr × P1 , (d, 1)). However, this is clear for suppose that (µ : C → Pr × P1 , {pi}) and (µ′ : C ′ → Pr × P1 , {p′i}) are isomorphic. Then we have an isomorphism f : C → C ′ that satisfies µ′ ◦ f = µ, and sends the marked points to each other. The following diagram shows that g · µ and g · µ′ are isomorphic and hence in the same equivalence class. C

f

′ .................................................................................................................................... ..... . ....... .... .. ... ... ... ... . . . . ... ..... ... .... ′ .......... ...... ... .... .. ... .... .... .. ... ... ... . . . . . .... . ... ... .... ... ... ... .... ... .... ... ... .. .... ... .... .. ... . . ... . . . . . . . ... ......... ... ...... ... ... ... ... r 1 ...... ... ′ ... ... ... ... .... ... ... ... ... ... ... ... . ... ... ... ... ... ... ... ... ... ... .. ... . ... .. ... ... .... .... ......... ........ ......... . . .

C

µ

µ

g·µ P ×P

g·µ

id × g

Pr × P1 This can also be thought of as a special case of Proposition 2.2.3, taking g : Pr × P1 → Pr × P1 .

31

Now that we have G acting on our two compactifications, we would like to see how these actions are related. Lemma 2.3.2. The map from 2.2.2 φ : M 0,n (Pr × P1 , (d, 1)) → (P1 )n × Prd is equivariant with respect to the above G actions. Proof. We show that both the evaulation morphisms evi and the Givental ϕ map are equivariant. Take a point in M 0,n (Pr × P1 , (d, 1)). Choose a representative (µ : C → Pr × P1 , {pi }) such that µ restricted to C0 is the graph of r + 1 degree d′ forms (f1 , . . . , fr ). Write C = C0 ∪ Ti as in Proposition 2.2.1, such that Ti ∩ C0 = qi . Also write µi = π1 ◦ µ|Ti . If we look at the image of the above map under the Givental map, we see that ϕ(µ) will be the product of r + 1 forms (f0 , · · · fr ) of degree d′ with no common zero that µ|C0 represents, along with a form h of degree d − d′ that vanishes at the pi with the correct degrees. Looking at the action of g, we see that g · ϕ(µ) will be the product of (f0 · g −1, . . . , fr · g −1) which are r + 1 forms of degree d′ with no common zero with a form h′ of degree d − d′ that vanishes at g(qi ) with the same degree that h vanished at qi . We now need to calculate ϕ(g · µ). With the above notation, we see that g · (µ : C → Pr × P1 ) would send p ∈ Ti to (µi (p), g(qi)), and p ∈ C0 to (µ0 (p), g(p)). We now choose a representation of the map g · (µ : C → Pr × P1 , {pi}) as in Proposition 2.2.1. Take the curve C ′ = g(C0) ∪ Ti , where now the teeth

32

Ti are glued to C0 at g(qi ). Define the map from C ′ → Pr × P1 as (µ0 ◦ g −1, id) on g(C0 ), and agrees with µi on the other teeth. We look at the image under the Givental map. The image will be the r + 1 degree d′ forms µ0 ◦ g −1, along with a form h that vanishes at g(qi) of the correct degree. This is the same as g · ϕ(µ). This shows that ϕ is equivariant. That the evaluation morphisms are equivariant is immediate. 2.3.2

Chamber Decomposition of PicG ((P1 )n × Prd )

Immediately, one would expect that M 0,n (Pr , d) is the quotient of M 0,n (Pr × P1 , (d, 1)) by G as G “takes into account” the map to P1 . The question is, “How do we take the quotient?” If we morally think of the quotient as forgetting the map to P1 , we see that there are 2 + 2n configurations in M 0,n (Pr × P1 , (d, 1)) that map to the open locus M0,n (Pr , d) ⊂ M 0,0 (Pr , d) corresponding to maps with smooth domain. In fact we can say more. Proposition 2.3.3. Let [µ : C → Pr , {pi}] ∈ M 0,n (Pr , d) be a stable map such that the domain curve has k components. Then there are 3k − 1 + 2n components in M 0,n (Pr × P1 , (d, 1)) that map to [µ] under the forgetful map f : M 0,n (Pr × P1 , (d, 1)) → M 0,n (Pr , d). Proof. This is immediate when you consider where we could attach a map to P1 . For each of the k components, we could attach a new genus 0 curve and a bi-degree (0, 1) map to it. This component would be collapsed by the map f . For each of the n points, we could attach a new genus-0 curve and a 33

bi-degree (0, 1) map to it with the marked on it on it. This component would be collapsed by f . Also for each of the n points we can add a degree (0, 0) map with the marked point on it. This (0, 0) component will also intersect another degree (0, 1) component in a second point. For each of the k −1 nodes, we could replace the node with a genus 0 curve and a bi-degree (0, 1) map. This would also be collapsed by the map f . Finally, for each of the the k components, we can replace the degree dk map to Pr with a bi-degree (dk , 1) map to Pr × P1 . (0,1)

(0,1)

(0,1)

(d,1) (d,0)

(d,0)

(d,0)

d

Figure 2.6: Components in M 0,1 (Pr × P1 , (d, 1)) Mapping to M0,1 (Pr , d) We deal with this by using GIT to find an open set in G such that the forgetful map is well behaved on this open set. All background concerning GIT will be taken from [3] and [22], though we recall the main Theorem of GIT here for future reference. Theorem 2.3.4. [3] [22] Let X be an algebraic variety, L a G-linearized line bundle on X. Then there are open sets X s (L) ⊂ X ss (L) ⊂ X (the stable and semi-stable loci), such that the quotient π : X ss (L) → X ss (L)/G is quasi-projective and a “good categorical quotient”. This says (among other things) that for any other G-invariant morphism g : X ss (L) → Z, there is a 34

unique morphism h : X ss (L)//G → Z satisfying h ◦ π = g. If we restrict to X s (L) then we have a “geometric quotient”. This says (among other things) that the geometric fibers are orbits of the geometric points of X, and the regular functions on X s (L)/G are G -equivariant functions on X. Of special interest to us will be when X is proper over C (as in this case) and L is ample, as here X ss (L)/G will be projective [22]. A word of warning. We are using stable to mean two different things in this Thesis. One one hand, we talk about stable points with respect to the group action. On the other hand, we have stable maps, meaning maps with finite automorphism group. The usage should be clear from the context. Notice that a GIT quotient is not unique, but rather depends upon a choice of a G-linearized line bundle. Definition 2.3.1. [3] [22] A G-linearization of L is an action σ¯ : G × L → L such that • the diagram σ ¯

G × L −−−→ L     πy id×π y σ

G × X −−−→ X

is commutative,

• the zero section of L is G- invariant And, while in general there will be a choice of linearization to be made (for example see the coming pages), at certain times there are no choices. For example: 35

Proposition 2.3.5. Let G act on a normal, irreduciable, proper variety X (such as Pr ). Then any line bundle admits a unique G linearization. Proof. By Theorem 7.2 from [3], there is an exact sequence α

δ

0 −−−→ Ker(α) −−−→ PicG (X) −−−→ Pic(X) −−−→ Pic(G) where α : PicG (X) → Pic(X) forgets the linearization. However, by our assumptions, we have Pic(G) = 0 and Ker(α) ∼ = χ(G), the group of characters of G. However, G doesn’t have any characters, and so α is an isomorphism. Since X ss (L) = X ss (L⊗n ) for any n > 0, [3] we see that all ample line bundles give the same stable locus for an action on Pr . However since we are dealing with a product of projective spaces, there is more than once choice of ample line bundle on (P1 )n × Prd . Proposition 2.3.6. PicG ((P1 )n × Prd ) ∼ = Zn+1 Proof. For any vector ~k = (k1 , · · · , kn , kn+1) ∈ Zn+1 , we define a line bundle on (P1 )n × Prd by L~k =

n+1 O

πi∗ (O(ki ))

i=1

where πi is projection onto the i-th component. Every line bundle on (P1 )n ×Prd is isomorphic to L~k for some choice of ~k ([12]). We need only show that each of these line bundles has one (and only one) linearization. However, since each πi is G-equivariant, and each of the restrictions of L~k to a factor has a unique linearization ([3]), L~k has a canonical G - linearization. 36

Corollary 2.3.7. L~k is ample ⇐⇒ ki > 0 Proof. If all ki > 0 then L~k defines the projective embedding Veronese

(P1 )n × Prd −−−−−→

Qn

i=1

P(1+ki )−1 × P(

rd+r+d+kn+1 rd+r+d

−−−→ P(( Segre

)−1

Qn

i=1

1+ki )×(

rd+r+d+kn+1 rd+r+d

))−1

On the other hand if some multiple of L~k defines a closed embedding, restricting it to any factor will be ample. But this is OP1 (ki ) (or OPrd (kn+1 )) and these are ample iff ki , kn+1 > 0. In order to find the stable and semi-stable loci in (P1 )n × Prd , we will in fact look at the image under the above Veronese / Segre maps. The main point is the following. Proposition 2.3.8. Let Ω be the composition of the Veronese and Segre maps above. Then 1 n

((P ) ×

Prd )ss (L~k )



= Ω { P (( −1

Qn

i=1

1+ki )×(

rd+r+d+kn+1 rd+r+d

))−1

ss

(O(1))}

and similarly for the stable locus. Proof. Call the image projective space PBIG . There is an action of G on PBIG such that Ω is G-equivariant. First, we show that there is an action of G on Pk such that the Veronese map P1 → Pk is G equivariant. We need a reparesentation of G in GL(k + 1). We can explicitly write it out, where we

37

choose [x, y] as coordinates on P1 and the obvious coordinates [xk : xk−1 y : · · · : y k ] on Pk . Namely,

where ai,j =

Pj

n=0

ρ : G → GL(k + 1)   a b → [ai,j ]ki,j=0 c d  i  n k−i−n j−n i−j+n k−i b a d c . This is the coefficient of n j−n

xk−j y j in (ax + by)k−i(cx + dy)i, and is a homomorphism. We can also define  n+1 the representation of G into GL( rd+r+d+k ) in a similar way. We now have rd+r+d

representations

ρi : G → GL(ki + 1) and

ρn+1

  rd + r + d + kn+1 ). : G → GL( rd + r + d

We define the action on PBIG by taking the tensor representation. Namely, for x ∈ V1 and y ∈ V2 , we define g · (x ⊗ y) = g · x ⊗ g · y. This extends to an action on all of PBIG . This action makes Ω G invariant by definition. Now look at the composition (P1 )n × Prd → Ω((P1 )n × Prd ) ֒→ PBIG . We apply the following Theorem of [22] to each of these arrows. Theorem 2.3.9. [22] (pg 46) Assume that f : X → Y is finite, G- equivariant with respect to actions of G on X and Y . If X is proper over k (C for us) and M is ample on Y , then X ss (f ∗ M) = f −1 {Y ss (M)} and the same result holds for the stable locus. Finally, that Ω∗ O(1) = L~k is obvious. We are now able to determine the stable and semi-stable locus in the linear-sigma model. 38

Theorem 2.3.10. Let ~k = (k1 , k2 , . . . , kn+1) ∈ Zn+1 + . Then [x1 : y1 ]×· · ·×[xn : yn ] × [aji ] ∈ ((P1 )n × Prd )ss (L~k ) (respectively ∈ ((P1 )n × Prd )s (L~k )) if for every point p ∈ P1 X

1 ≤ 2

ki + dp · kn+1

i | [xi :yi ]=p

n X

ki + d · kn+1

i=1

!

(respectively strict inequality holds) where dp is the degree of common vanishing of the forms f0 , . . . , fr at p ∈ P1 . Proof. Let T be the maximal torus of SL2 (C), equal to the image of the 1parameter subgroup

 −1  t 0 . λ(t) = 0 t

We choose coordinates aji on Prd , where aji is the coefficient of xd−i y i in fj (x, y). Similarly, we choose the following coordinates on PBIG . For 0 ≤ P P si ≤ ki (1 ≤ i ≤ n), and vij such that di=0 rj=0 vij = kn+1 , we have the coordinate xki i −si yisi (aji )vij . Then T acts on PBIG by Pn

λ(t) · (xki i −si yisi (aji )vij ) → t(

i=1

P 2si −ki )+( ij (d−2i)vij ) ki −si si j vij xi yi (ai )

By the above Lemma, we know that it’s enough to compute the semistable locus of this action on PBIG and pull it back via the various inclusions and embeddings. Luckily we know how to compute the semi-stable locus of a torus acting on a projective space. From [3] we know that a point of projective space is stable (resp semi-stable) with respect to T if and only if 0 ∈ interior(wt) ( resp 0 ∈ wt). In our case, the weight set (wt) is the subset of {−

n X

ki − d · kn+1 , . . . ,

i=1

n X i=1

39

ki + d · kn+1 }

consisting of powers of t such that the coordinate xki i −si yisi (aji )vij is non zero. P P If the point is unstable, then all the powers ( ni=1 2si − ki ) + ( ij (d −

2i)vij ) < 0 (or perhaps all > 0.) So xki i −si yisi (aji )vij = 0 if 0≤(

n X i=1

X 2si − ki ) + ( (d − 2i)vij ) ⇐⇒ ij

0≤2

n X

(si − ki ) − 2

i=1

X

i · vij +

ij

X

n X

ki + dkn+1 ⇐⇒

i=1

n X

i · vij +

(ki − si ) ≤

i=1

ij

1 2

n X

!

ki + dkn+1 .

i=1

Define the following sets in (P1 )n × Prd : US = {[x1 : y1 ] × · · · × [xn : yn ] × [aji ] | n X X 1 xki i −si yisi (aji )vij = 0 if i · vij + (ki − si ) ≤ 2 ij i=1

n X

!

ki + dkn+1 }

i=1

and X = {[x1 : y1 ] × · · · × [xn : yn ] × [aji ] | ! n 1 X ki + dkn+1 < 2 i=1

X

ki + kn+1 · d[1:0] }

[xi :yi ]=[1:0]

We show that US = X First, assume that X ⊂ US. Let x = {[x1 : y1 ] × · · · × [xn : yn ] × [aji ]} be in US \ X. So, xki i −si yisi (aji )vij = 0 if X

i · vij +

ij

n X

(ki − si ) ≤

i=1

1 2

n X

!

ki + dkn+1 }.

i=1

But we also have X

[xi :yi ]=[1:0]

ki + kn+1 · d[1:0]

1 ≤ 2

40

n X i=1

!

ki + dkn+1 .

Then, take si = 0 if [xi : yi ] = [1 : 0]. And at least one of the ajd[1:0] 6= 0. For that value of j, let vij = kn+1 . Then we have n X i=0

(ki − si ) +

X

X

i · vij =

ij

ki + kn+1 · d[1:0]

[xi :yi ]=[1:0]

1 ≤ 2

n X

ki + dkn+1

i=1

!

.

The coordinate xki i −si yisi (aji )vij 6= 0 by construction, which says x ∈ / US, a contradiction. Thus US ⊆ X. Now, assume that US ⊂ X Take y = {[x1 : y1 ] × · · · × [xn : yn ] × [aji ]} ∈ X \ US. So xki i −si yisi (aji )vij 6= 0, but X ij

n X 1 i · vij + (ki − si )} ≤ 2 i=1

n X

ki + dkn+1

i=1

!

Combining with the fact that y ∈ X, we see that n X

(ki − si ) +

i=1

X

X

i · vij
0, we get the following Lemma.

41

Lemma 2.3.11. [x1 : y1 ] × · · · × [xn , yn ] × [aji ] is unstable with respect to T if ! n X 1 X ki + dkn+1 ki + kn+1 · d[1:0] > 2 i=1 i | [xi :yi ]=[1:0]

or

X

ki + kn+1 · d[0:1]

i | [xi :yi ]=[0:1]

1 > 2

n X

ki + dkn+1

i=1

!

We are now ready to move onto stability with respect to G. Suppose that [x1 : y1 ] × · · · × [xn , yn ] × [aji ] is stable with respect to G and there is a point p in P1 such that X

[xi :yi ]=p

1 ki + kn+1 · dp > 2

n X

!

ki + dkn+1 .

i=1

Let g ∈ G map p → [1 : 0]. Then g · [x1 : y1 ] × · · · × [xn , yn ] × [aji ] is unstable with respect to T, and [x1 : y1 ] × · · · × [xn , yn ] × [aji ] is unstable with respect to G, contradicting the assumption. Now assume that [x1 : y1 ] × · · · × [xn , yn ] × [aji ] is unstable, but has no point p such that X

i|[xi :yi ]=p

1 ki + kn+1 · dp > 2

n X

!

ki + dkn+1 .

i=1

Then, there is some maximal torus T ′ with which [x1 : y1 ] × · · · × [xn , yn ] × [as ] is unstable. For any maximal torus in G, there is g ∈ G such that gT ′g −1 = T . Then we have that g · [x1 : y1 ] × · · · × [xn , yn ] × [aji ] is unstable with respect to T , hence must have either [1 : 0] or [0 : 1] satisfiying Lemma 2.3.11. Then [x1 : y1 ] × · · · × [xn , yn ] × [aji ] has g −1 [1 : 0] satisfying Lemma 2.3.11. Corollary 2.3.12. 1 ((P1 )n × Prd )ss (L~k ) 6= ∅ ⇐⇒ ∀i ∈ (1, . . . , n), ki ≤ 2 42

n X i=1

ki + d · kn+1

!

1 ((P1 )n × Prd )s (L~k ) 6= ∅ ⇐⇒ ∀i ∈ (1, . . . , n), ki < 2

n X

ki + d · kn+1

i=1

!

The next three Corollaries give line bundles such that the stable locus equals the semi-stable locus for all different parity combinations of n and d. This will be helpful when we need to do the actual calculations in our Main Theorem. Corollary 2.3.13. Suppose that ~k = (1, 1, . . . , 1). Then n + d = odd ⇐⇒ ((P1 )n × Prd )ss (L~k ) = ((P1 )n × Prd )s (L~k ) Corollary 2.3.14. Suppose that ~k = (1, 1, . . . , 1, 2). Then n + d = even , n odd ⇒ ((P1 )n × Prd )ss (L~k ) = ((P1 )n × Prd )s (L~k ) Corollary 2.3.15. Suppose that ~k = (1, 2, 2, . . . , 2, 1). Then n + d = even , n even ⇒ ((P1 )n × Prd )ss (L~k ) = ((P1 )n × Prd )s (L~k ) When kn+1 → 0, we get the familiar case of points on P1 from the study of Fulton-MacPherson spaces and M 0,n . Corollary 2.3.16. Suppose that ((P1 )n × Prd )ss (L~k ) = ((P1 )n × Prd )s (L~k ). If for any I ⊂ (1, 2, . . . , n) X i∈I

1 ki < 2

n X i=1

ki + d · kn+1

!

=⇒

X i∈I

ki + d · kn+1

1 < 2

n X i=1

ki + d · kn+1

!

then ((P1 )n × Prd )s (L~k ) = ((P1 )n )s (Lk1 ,...,kn ) × Prd where the stable locus on the right is the usual stable locus for configurations of points on P1 . See [3] for details. 43

Similarly, we can look in the other direction, and see what happens when k1 , ..., kn → 0. Corollary 2.3.17. Suppose that ((P1 )n × Prd )ss (L~k ) = ((P1 )n × Prd )s (L~k ). If ⌊d/2⌋ · kn+1

n X

1 < 2

ki + d · kn+1

i=1

n X

!

=⇒

ki + ⌊d/2⌋ · kn+1

i=1

1 < 2

n X i=1

ki + d · kn+1

!

then ((P1 )n × Prd )s (L~k ) = (P1 )n × (Prd )s (Lkn+1 ) We are now ready to describe the chamber decomposition of the ample cone of PicG ((P1 )n × Prd ). As a first step we normalize our line bundle so that we form the simplex ∆ = {(k1 , k2 , . . . , kn+1)|

n X

ki + d · kn+1 = 2}

i=1

Then for each subset I ∈ (1, 2, . . . , n) and each integer 0 ≤ dI ≤ d, we get a wall WI,dI given by X

ki + dI · kn+1 = 1

i∈I

and the walls break ∆ into chambers. Following [14], we mention the following obvious statements. 1. WS,dS = WS c ,d−dS

44

2. Each interrior wall divides ∆ into two parts {(k1 , k2 , . . . , kn+1)|

X

ki + dI · kn+1 ≤ 1}

i∈I

and {(k1 , k2 , . . . , kn+1)|

X

ki + dI · kn+1 ≥ 1}

i∈I

′ 3. Two vectors ~k = (k1 , . . . , kn+1) and k~′ = (k1′ , . . . , kn+1 ) lie in the same

chamber if for all I ⊂ {1, 2, . . . } and 0 ≤ dI ≤ d then X

ki + dI · kn+1 ≤ 1 ⇐⇒

i∈I

X

′ ki′ + dI · kn+1 ≤1

i∈I

This means that vectors in the same chamber will define the same stable and semi-stable loci, and hence the same quotient. 4. There are semi-stable points that aren’t stable iff ~k lies on a wall. Recall that our goal isn’t to take the GIT quotient of (P1 )n × Prd , but of M 0,n (Pr × P1 , (d, 1)), and at this point we haven’t said anything about the stable or semi-stable loci in M 0,n (Pr × P1 , (d, 1)). We are able to pull back the stable locus via φ, by the following Theorem of Yi Hu. Theorem 2.3.18. [13] Let π : Y → X be a G-equivariant projective morphism between two (possibly singular) quasi-projective varieties. Given any linearized ample line bundle L on X, choose a relatively ample linearized line bundle M on Y . Assume moreover that X ss (L) = X s (L). Then there exists a n0 such that when n ≥ n0 , we have Y ss (π ∗ Ln ⊗ M) = Y s (π ∗ Ln ⊗ M) = π −1 {X s (L)} 45

For example, the locus of maps in M 0,0 (Pr × P1 , (d, 1)) that are stable will be maps such that no tooth of the comb C has degree ≥ d/2. See Figure 2.3.2 to see how the stable locus depends on the line bundle. 3

3

2 (1,0)

1

(1,0)

(1,1)

2 (1,1) 1

Unstable for (1,1,1,1)

Stable for (1,1,1,1)

3 2

3

(1,0)

(1,0)

2 1 (1,1)

1

Unstable for (1,1,3,1)

(1,1)

Unstable for (1,1,3,1)

Figure 2.7: Stable Locus of M 0,3 (Pr × P1 , (2, 1))

2.3.3

Main Theorem

We are now ready to present our GIT description of M 0,n (Pr , d). The construction is similar to that of M 0,n from [15]. We state it similarly. First let E 46

be an effective divisor with support the full exceptional locus of φ, such that −E is φ ample. Such an E exists by the following Lemma from [20]. Lemma 2.3.19. [20] (pg. 70) Let f : X → Y be a birational morphism. Assume that Y is projective and X is Q-factorial. Then there is an effective f -exceptional divisor E such that −E is f -ample. Theorem 2.3.20. For each linearized line bundle L ∈ PicG ((P1 )n × Prd ) such that ((P1 )n × Prd )ss (L) = ((P1 )n × Prd )s (L) 6= ∅ and for each sufficiently small ǫ > 0, the line bundle L′ = φ∗ (L)(−ǫE) is ample and (M 0,n (Pr × P1 , (d, 1)))ss(L′ ) = (M 0,n (Pr × P1 , (d, 1)))s(L′ ) = φ−1 {((P1 )n × Prd )ss (L)}. There is a canonical identification (M 0,n (Pr × P1 , (d, 1)))s (L′ )/G = M 0,n (Pr , d) and a commutative diagram f

(M 0,n (Pr × P1 , (d, 1)))ss(L′ ) −−−→   φy ((P1 )n × Prd )ss (L)

M 0,n (Pr , d)   y

−−−→ ((P1 )n × Prd )ss (L)/G

where φ is the generalized Givental map, f is the forgetful morphism. Proof. For the first two statements, we apply the above Theorem 2.3.18 of Hu. Following the notation from [15], let U be the semi-stable locus in (P1 )n × Prd 47

for the above action of G corresponding to L~k . Recall that this corresponds to ([xi , yi], f0 , . . . fr ) such that for any p ∈ P1 , we have X

i|[xi :yi ]=p

1 ki + kn+1 · dp ≤ 2

n X

!

ki + kn+1 · d .

i=1

Let U ′ = φ−1 (U). Let the corresponding quotients be Q and Q′ . We have the obvious composition of G invariant maps: U ′ → U → Q. And by the universal properties of GIT quotients, we get a proper birational map Q′ → Q. Similarly, since G acts trivially on M 0,n (Pr , d), we have by the universal property again a proper birational map from Q′ → M 0,n (Pr , d). We will show that this is an isomorphism by showing that both sides have the same Picard number. This is enough since both sides are Q-factorial by Proposition 2.1.2. We have: ρ(Q′ ) = ρ(U ′ ) = ρ(U) + e(U) = ρ(Q) + e(U) where e(u) is the number of φ exceptional divisors that meet U ′ . Since φ is an isomorphism on the open locus M0,0 (Pr × P1 , (d, 1)) ⊂ M 0,0 (Pr × P1 , (d, 1)), we need only look at the boundary divisors in M 0,0 (Pr × P1 , (d, 1)). We use the following Lemma and it’s Corollary to see what divisors are φ-exceptional Lemma 2.3.21. φ(D(N1, N2 , d1 , d2)) ⊂ (P1 )n × Prd has codimension |N2 | + (r + 1)d2 − 1.

48

Proof. The idea for this proof comes from Kirwan’s book [19]. First notice that φ(D(N1 , N2 , d1 , d2 )) = (p1 , p2 , . . . , pn , f0 , . . . , fr ) where pi = pj for i, j ∈ N2 , and each of the fs vanish at that point of multiplicity d2 . First, we calculate the codimension of (p1 , p2 , . . . , pn , f0 , . . . , fr ) where each of the pi is [0 : 1], and where each of fj has a zero of order d2 at [0 : 1] and an order of d −d2 = d1 at [1 : 0] (i.e. each fj consists of only the monomial xd2 y d1 ). It is clear this has codimension n + rd + r + d − r = n + (r + 1)d. If we remove the condition that each fj have a root of order d1 at [1 : 0], then we allow each fj to have higher powers of x. We also remove the condition that those pi with i ∈ / N2 are equal to [0 : 1]. Thus we see that the the set of (p1 , p2 , . . . , pn , f0 , . . . , fr ) such that [0 : 1] = pi for i ∈ N2 , and each fi vanishes at [0 : 1] with multiplicity d2 has codimension n + (r + 1)d − (r + 1)d1 − |N1 | = |N2 | + (r + 1)d2 . Finally, we act on this set by G. We subtract one from the above codimension because G has dimension two, but we don’t count the two dimensional stabilizer of [0 : 1]. Corollary 2.3.22. Every bounadry divisor D(N1 , N2 , d1 , d2) of M 0,n (Pr × P1 , (d, 1)) is φ-exceptional unless 1. |N2 | = 2 and d2 = 0 2. r = 1, |N2| = 0 and d2 = 1. 49

Case 2 in the above Corollary reflects that two polynomials have a common root iff the resultant vanishes. Next we show that ρ(Q′ ) is independent of the chamber from where L~k comes from. We check that as we cross a wall WI,dI , ρ(Q′ ) doesn’t change. Let our two open sets be U1 and U2 . Recall that WI,dI breaks our chamber into two parts {(k1 , . . . , kn , kn+1 )|

X

ki + dI · kn+1 ≤ 1}

i∈I

and {(k1 , . . . , kn , kn+1 )|

X

ki + dI · kn+1 ≥ 1}

i∈I

so suppose that U1 meets the first set. Notice that U1′ and U2′ meet the same divisors D(N1 , N2 , d1 , d2 ) except that U1′ meets D(I c , I, d − dI , dI ) but not D(I, I c , dI , d − dI ). Similarly, U2′ meets D(I, I c , dI , d − dI ) but not D(I c , I, d − dI , dI ). If 2 < |I|, 1 < r and 1 ≤ dI ≤ d, or r = 1 and 1 < di ≤ d, then Q1 99K Q2 is a small modification (an isomorphism in codimension 1 in the notation of [20]). Hence ρ(Q1 ) = ρ(Q2 ), and it’s clear that e(U1 ) = e(U2 ). If 2 = |I| and dI = 0, (or r = 1, |I| = 0, dI = 1), then we see that Q1 99K Q2 contracts the divisor (p1 , . . . , pn , f0 , . . . fr ) where pi = pj , i, j ∈ I ( or f0 , f1 have a common root.). Therefore ρ(Q1 ) = ρ(Q2 ) + 1. However, by Corollary 2.3.22, we see that the divisor D(I c , I, d, 0) with |I| = 2 lying over U1 is not exceptional, while it’s complement D(I, I c , 0, d) lying over U2 is exceptional. (Similarly for D(N, 0, d − 1, 1)). Hence e(U2 ) = e(U1 ) + 1. Putting these together we see that ρ(Q′1 ) = ρ(Q′2 ) as desired. 50

Finally, we prove the Theorem for one vector of one chamber. Here we look at all divisors D(N1 , N2 , d1 , d2) ∈ M 0,n (Pr × P1 , (d, 1)). We have 2n ways to distribute the n points on the domain curve, and we can label the collapsed component with any degree ≤ d. Hence there are 2n (d + 1) potential configurations, however the configurations D(I, I c , d, 0) are not stable maps if |I| = n or n − 1. Hence there are 2n (d + 1) − n − 1 total boundary divisors in M 0,n (Pr ×P1 , (d, 1)). We need to determine how many are stable (with respect to the group). We do several calculations, as a given linearization ~k may lie in a maximal chamber for certain values of d, n, but lie on a wall for others. All the calculations are very similar though. Assume that r > 1. CASE 1 (d + n odd. d > n) We choose the linearization corresponding to (1, 1, 1, . . . , 1, 1). We count the unstable divisors, i.e. the number of D(N1 , N2 , d1 , d2 ) such that d+n+1 2

|N2 | + d2 ≥ Any divisor D(N1 , N2 , d1 , d2 ) with

d+n+1 2

≤ d2 ≤ d is unstable. There are

) of these. Thus the total number of unstable divisors is 2n ( d−n+1 2 # with d2 = d+n−1 2

# with d2 = d+n−3

2 z }| {    {   }| n d−n+1 n n + (2n − )+... ) + (2n − − 2n 2 1 0 0

z

# with d2 = d−n+1 2

}| {      n n n −···− − + 2n − n−1 1 0 z

51

       n n n d−n+1 n −···−1 − (n − 1) + n2 − n =2 n−1 1 0 2         d−n+1 n n n n n =2 −···−1 − (n − 1) + n2 − n 1 n−1 n 2     n X d−n+1 n = 2n + n2n − (i) i 2 i=1   d−n+1 + n2n − n2n−1 = 2n−1 (d + 1) = 2n 2 n



Hence the total number of stable divisors is 2n−1 (d + 1) − 1 − n (stable with respect to the group). The number which are φ exceptional are all except  those where I c = 2 (by Corollary 2.3.22). So there are 2n−1 (d + 1) − 1 − n − n2 φ-exceptional divisors. Thus, since ρ(Q) = n + 1, ′

ρ(Q ) = ρ(Q) + e(U) = n + 1 + 2

n−1

    n n n−1 . = 2 (d + 1) − (d + 1) − 1 − n − 2 2

CASE 2 (d + n odd, d < n) We again choose the linearization corresponding to (1, 1, 1, . . . , 1, 1). We count the total number of un-stable divisors, i.e. D(N1 , N2 , d1 , d2 ) with |N2 | + d2 ≥

d+n+1 . 2

In this case the number of un-

stable divisors is # with d2 =d

# with d2 =d−1

}|     }|   { z   { n n n n n − · · · − n−d−1 + 2n − − · · · − n−d−1 − n−d+1 2n − 0 0 2 2 2 z

# with d2 =0

z

{    }| n n − · · · − n+d11 + · · · + 2n − 0 2 The first group above pairs with the last in order to sum to 2n+1 − 2n = 2n . Similarly for the second and the second to last. If d is odd, then the d + 1 groupings above pair up completely to get

d+1 n 2 2

= 2n−1 (d + 1). If d is even,

there there is a center sum that doesn’t pair with anything. However, this 52

central sum adds to 2n−1 . Thus there are again d2 2n + 2n−1 = 2n−1 (d + 1) unstable divisors. The number of stable divisors with respect to the group is 2n−1 (d + 1) − 1 − n. Again, the number which are φ exceptional are all except  those where I c = 2 (by Corollary 2.3.22). So there are 2n−1 (d + 1) − 1 − n − n2 φ-exceptional divisors. Thus ′

ρ(Q ) = ρ(Q) + e(U) = n + 1 + 2

n−1

    n n n−1 . = 2 (d + 1) − (d + 1) − 1 − n − 2 2

CASE 3 (d + n even, n odd) We choose the linearization corresponding to (1, 1, . . . , 1, 2). If d >

n+2d 2

we proceed as in CASE 1. If n >

n+2d 2

we

proceed as in CASE 2. CASE 4 (d + n even, n even) We choose the linearization corresponding to (1, 2, 2, . . . , 2, 1). If d > 2n+d−1 2

2n+d−1 , 2

then proceed as in CASE 1. If 2n − 1 >

proceed as in CASE 2. Note that care must be taken when n = 2 and d = 1, 2. For here,

ρ((P1 )n × Prd ) = 2 for the given linearizations (instead of the expected 3). This is because the unstable locus contains a divisor. However, we wouldn’t need to subtract out the divisor D(0, 2, d1, d2 ) for not being φ exceptional, because it would have been unstable w.r.t. the group. Thus, the sums work out to be the same. We have shown for every line bundle such that the stable locus equals  the semi-stable locus that ρ(Q′ ) = 2n−1(d + 1) − n2 . From [27] we have r

ρ(M 0,n (P , d)) = 2

n−1

which completes the proof for r > 1. 53

  n (d + 1) − 2

When r = 1 we repeat the above construction. Here we see, by Corollary 2.3.22, that the divisor D(N, 0, d − 1, 1) is not φ-exceptional. Thus we subtract one from the above count of φ-exceptional divisors. Thus     n n n−1 ′ n−1 −1 = 2 (d+1)− ρ(Q ) = ρ(Q)+e(U) = n+1+2 (d+1)−2−n− 2 2 An immediate consequence of Theorem 4.4 in [1] gives ρ(M 0,n (P1 , d)) and it agrees with the above calculation. In the case when r = 0, then d = 0 or else the moduli space is empty. Thus P00 = pt. The calculation follows as now we are only dealing with stable curves, and not stable maps and was proven originally in [15]. We have that   n ′ n−1 − 1 = ρ(M 0,n ) ρ(Q ) = 2 − 2

There are three immediate corollaries that are interesting. The first was proven by Keel and Hu in [15]. By letting d, r = 0 in the above Theorem we have. Corollary 2.3.23. [15] For each linearized line bundle L ∈ PicG ((P1 )n ) such that ((P1 )n )ss (L) = ((P1 )n )s (L) 6= ∅ and for each sufficiently small ǫ > 0, the line bundle L′ = φ∗ (L)(−E) is ample and (P1 [n])ss (L′ ) = (P1 [n])s (L′ ) = φ−1 ((P1 )n )s s(L) There is a canonical identification (P1 [n])s (L′ )/G = M 0,n 54

and a commutative diagram f

(P1 [n])ss (L′ ) −−−→   φy

M 0,n   y

((P1 )n )ss (L) −−−→ ((P1 )n )ss (L)/G)

Proof. Just recall from earlier that M 0,n ({Spec(C)} × P1 , (0, 1)) ∼ = P1 [n] and M 0,n ({Spec(C)}, 0) ∼ = M 0,n . Also the Fulton-MacPherson map P1 [n] → (P1 )n is exactly the product of evaluation morphisms φ. This unifies M 0,n (Pr , d), P1 [n], and M 0,n into one construction. Secondly, we find that the Grassmannian of lines is a GIT quotient of a projective space. Note that when n = 0, then the linear-sigma model consists of Prd ∼ = P(r+1)(d+1)−1 . There is only one ample line bundle (up to multiple) on this space, and it has a unique linearization. Hence, there are no choices made in taking the quotient. Corollary 2.3.24. The Grassmannian of lines in Pr is a GIT quotient of Pr1 = P2(r+1)−1 by the above action of G. Proof. By Example 2.1.5, we know that M 0,0 (Pr , 1) = M0,0 (Pr , 1) = G(1, r). By the Theorem (M 0,0 (Pr × P1 , (1, 1)))s/G = G(1, r). But M 0,0 (Pr × P1 , (1, 1))s = M0,0 (Pr × P1 , (1, 1)) ∼ = (Pr1 )s . Finally, the Theorem reduces questions about M 0,n (Pr , d) to questions about M 0,n (Pr × P1 , (d, 1)). For example, if we knew the Chow rings of 55

M 0,n (Pr × P1 , (d, 1)) we could compute those of M 0,n (Pr , d) by [31]. It also provides us with a birational morphism to a projective variety which we examine closer in the next chapter. Notice, by the above comment before 2.3.23, that this quotient is categorical (doesn’t depend on any choices). Corollary 2.3.25. There is a proper, birational morphism ϕ¯ : M 0,0 (Pr , d) → (Prd )s (O(1))/G

56

Chapter 3 The Betti Numbers of ((P1)n × Prd)ss/G

In this chapter we compute the the Betti numbers of the GIT quotient of the pointed linear-sigma model described above. The purpose is that we would hope that knowing the cohomology of this space, as well as the factorization of the birational map ϕ¯ : M 0,n (Pr , d) → (P1 )n × Prd )ss /G. that we will be able to compute the Chow Rings of M 0,n (Pr , d). Theorem 3.0.26. The Poincar´e polynomial of ((P1 )n × Prd )ss (L(1,1,...,1) )/G when d + n is odd, n > d is (1 + t2 )n (1 + t2 + · · · + t2(rd+r+d) )(1 − t4 )−1 −  n−1 d  X X n { t2(s+kr−1) (1 + t2 + · · · + t2r )(1 − t2 )−1 }− s − k d+n+1 k=0 s=

{

2

d+n d+n−s X  X s=n

k=0

 n t2(s+r(k+s−n)−1) (1 + t2 + · · · + t2r )(1 − t2 )−1 } n−k

57

When d > n, we have that the Poincar´e polynomial is (1 + t2 )n (1 + t2 + · · · + t2(rd+r+d) )(1 − t4 )−1 −  d−1 n  X X n { t2(s+r(s−n+k)−1) (1 + t2 + · · · + t2r )(1 − t2 )−1 }− n−k d+n+1 k=0 s=

{

2

d+n d+n−s X X  s=d

k=0

 n t2(s+r(s−n+k)−1) (1 + t2 + · · · + t2r )(1 − t2 )−1 } n−k

Proof. This is an application of a Theorem of Kirwan from [16]. We start by making some definitions. Recall from Chapter 2 that in this case, we have a representation ρ : G → GL(2n(rd + r + d + 1)). We look at the image of the torus  −1  t 0 0 t For I ⊂ {1, 2, . . . , n}, we have coordinates xI c yI aji . Then t acts with weights t · (xI c yI aji ) → t|I|−|I

c |+d−2i

xI c yI aji .

Thus our weight set is then a subset of {d + n, d + n − 2, . . . , 1, −1, . . . , −d − n}. The indexing set of our stratification will be all the positive weights, i.e. B = {(2s − n − d) |

d+n < s ≤ d + n} 2

For any s in this range (corresponding to a β ∈ B) we define the set Zs = {xI c yI aji | xI c yI aji = 0 if |I| − |I c | + d − 2i 6= s}. 58

Recall that we can rewrite under the embedding Ω Zs ∼ = {[x1 : y1 ] × · · · × [xn , yn ] × (f0 , . . . , fr ) | (#[xi : yi ] = [0 : 1])+d[0:1] = s and (#[xi : yi] = [1 : 0])+d[1:0] = n+d−s}. We then define Ys = {xI c yI aji | xI c yI aji = 0 if |I| − |I c | + d − 2i < s and at least one coordinate with |I| − |I c | + d − 2i = s is nonzero} In other words Ys ∼ = {[x1 : y1 ] × · · · × [xn , yn ] × (f0 , . . . , fr ) | (#[xi : yi ] = [0 : 1]) + d[0:1] = s} Finally, we define Ws = G · Ys . Thus Ws corresponds to ([xi , yj ], f0 , . . . fr ) such that there is a p ∈ P1 , such that X

1 + dp = s.

i|[xi :yi ]=p

Recall by Lemma 2.3.21, that Ws has components of different dimensions. Thus, let Ws,m be the component of codimension m, and let Ys,m and Zs,m be the obvious components of their respective spaces. Applying Theorem 8.12 from [16] gives the Poincar´e Polynomial as Pt ((P1 )n × Prd )ss (L(1,1,...,1) )/G) = Pt ((P1 )n × Prd ))Pt (BSU(2)) −

X s,m

59

t2Codim(Ws,m ) Pt (Zs,m)Pt (BS 1 )

And the theorem is obtained by plugging in the values for the Pt above and counting components. For example, when s = d + n, then Ws has only one component. It’s the locus of where all the points agree and all the forms vanish at that point. This has codimension s + rd − 1, and the corresponding Zs = {[0 : 1] × [0 : 1] × · · · × [0 : 1] × (a00 xd , a10 xd , . . . , ar0 xd )} is a Pr ֒→ (P1 )n × Prd which we know the Poincar´e polynomial for. When s = d+n−1, then there are two components of Wd+n−1 . Namely 1. Ws,s+r(d−1)−1 = { all the points agree and the forms vanish at that point with muliplicity d − 1} 2. Ws,s+rd−1 = {n−1 of the points agree and the forms vanish at that point with muliplicity d} Notice that there are



n n−1

components of Ws,s+rd−1 corresponding to which

of the n − 1 points agree. The corresponding Zs are 1. Zs,s+r(d−1)−1 = {[0 : 1] × · · · × [0 : 1] × (a01 xd−1 y, a11 xd−1 y, . . . , ar1 xd−1 y)} 2. Zs,s+rd−1 = {[1 : 0] × [0 : 1] × . . . × [0 : 1] × (a00 xd , a10 xd , . . . , ar0 xd )} and both of these are easily seen to again be copies of Pr ֒→ (P1 )n × Prd . We continue in this manner for all values of s in our indexing set.

60

Notice that we could repeat this entire process for any linearization (not just (1, 1, . . . , 1)) in exactly the same way. We conclude with a few examples. Example 3.0.1. In the case when n = 0, then we find the Betti numbers for the Grassmannians G(1, r) as expected. r 3 5 7

Pt ((Pr1 )ss /G) (1,1,2,1,1) (1,1,2,2,3,2,2,1,1) (1,1,2,2,3,3,4,3,3,2,2,1,1)

Table 3.1: Betti Numbers of (Pr1 )ss /G

Example 3.0.2. In [8] the Betti numbers of M 0,n (Pr , d) are calculated. We compare them here r 1 2 3

Pt (M 0,0 (Pr , 3)) (1,1,2,1,1) (1,2,5,7,9,7,5,2,1) (1,2,6,10,17,20,24,20,27,10,6,2,1)

Pt ((Pr3 )ss /G) (1,1,2,1,1) (1,1,2,2,3,2,2,1,1) (1,1,2,2,3,3,4,3,3,2,2,1,1)

Table 3.2: Comparison of Betti Numbers of M 0,0 (Pr , 3) and (Pr3 )ss /G

Example 3.0.3. The Poincar´e polynomial for ((P1 )2 × P21 )ss /G is 1 + 2t2 + 3t4 + 2t6 + t8 . Notice here that the dimension of the Picard group is 2, which is one less than the expected 3. This is because the locus where both marked points are the same is a divisor and is unstable (recall Chapter 2). We compare that with Pt (((P1 )3 × P22 )ss /G) = 1 + 4t2 + 7t4 + 10t6 + 10t8 + 10t10 + 7t12 + 4t14 + t16 61

which does have the expected dimension of the Picard group. Notice that all the examples obey Poincar´e duality, as they would have to.

62

Chapter 4 Intermediate Moduli Spaces

In this chapter we restrict ourselves to the case of n = 0, and make special use of Corollary 2.3.25, providing us with the map M 0,0 (Pr , d) → Prd /G. Recall that in the construction of M 0,0 (Pr , d) from [5], Fulton and Pandhairpande chose a basis t¯ = (t0 , . . . tr ) of H 0 (Pr , O(1)) and then constructed a smooth space M (t¯) that was a torus bundle over a certain “balanced” subset of M 0,d(r+1) . That is: ∗ r

r+1

(C ) (Sd ) M 0,d(r+1) ⊃ B ←−−− M (t¯) −−−−→ Ut¯ ⊂ M 0,0 (Pr , d)

Morally, we think of the quotient on the right as an ordering of the zeros of the map [f ] ∈ M 0,0 (Pr , d). The bundle on the left forgets f and only records the ordered “zeros” of it. The torus (C∗ )r says that if we know d zeros of a degree d form, that we know the form itself up to multiplying the entire form by an element of C∗ . From Corollary 2.3.23 or [15], we know that M 0,d(r+1) is the GIT quotient of P1 [d(r + 1)] where P1 [d(r + 1)] is the Fulton MacPherson compactification of d(r + 1) points on P1 . The stable set was pulled back from the stable set for the usual diagonal action on (P1 )d(r+1) . (See, for example,

63

[22].) In Theorem 2.3.20, we constructed M 0,0 (Pr , d) as a GIT quotient of M 0,0 (Pr × P1 , (d, 1)). These spaces are related by the above diagram. Namely, we have Figure 4.1, with each side commuting over their respective stable loci. P1 [d(r + 1)] ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ......... ..

M 0,0 (Pr × P1 , (d, 1))

.... ... .... ... .... .... .... ... .... .... ... .... ...... ......

(C∗ )r M 0,d(r+1) ........................................................................................................ M (t¯)

F-M

(P1 )d(r+1) .... .... .... .... .... .... .... .... .... .... .... ...... ......

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ......... ..

... ... .... ... ... ... .... ... .... ... ... .... ... .... ... ... ... .... ... .... ... ... ..... ... ........ r+1 ... d ...... ................................................................................................. ... 0,0 .... ... ... ... ... ... ... ... ... ... . ........ ..

(S )

ϕ

M

Prd

(Pr , d) ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ........ ..

ϕ¯

... .... .... .... .... .... .... .... .... .... .... ...... ......

(P1 )d(r+1) G

Prd G

Figure 4.1: Relating GIT Quotients Moving from the back to the front of the diagram corresponds to quotient by G. We hope to fill in the center of Figure 4.1 with rigid spaces (analgous to M (t¯)) such that all the squares commute. We will see (Remark 4.2.1) that this is impossible because the stable loci on the left and right don’t correspond to each other under the bundle and quotient maps in the center. However, we will be able to partially fill in the figure by constructing rigid versions of the spaces on the right that are C∗ bundles over some of the spaces on the left. After this, we factor the map from Corollary 2.3.25 ϕ¯ : M 0,0 (Pr , d) → (Prd )ss /G into a sequence of intermediate moduli-spaces. In [7], the FultonMacPherson map on the left is factored in such as way. Mustat¸ˇa in [23] uses 64

this to factor the evaluation morhpism e : M 0,1 (Pr , d) → Pr and we will make use of this.

Rigid Versions of M 0,0(Pr × P1 , (d, 1)) and Prd

4.1

Here, we fill in Figure 4.1, by constructing rigid covers analogous to M (t¯). We construct a cover G(t¯) → Vt¯ ⊂ M 0,0 (Pr × P1 , (d, 1)) and then describe Prd (t¯) → Wt¯ ⊂ Prd that was constructed by Mustat¸ˇa in [23]. Finally, we will relate these spaces via a GIT construction similar to Theorem 2.3.20. Constructing M 0,0 (Pr × P1 , (d, 1))

4.1.1

In this section we outline the construction of M 0,0 (Pr × P1 , (d, 1)). During the proof, we find our rigid cover G(t¯) of M 0,0 (Pr × P1 , (d, 1)). We begin with a definition similar to Definition 3.2 in [5]. Definition 4.1.1. Let t¯ = (t0 , t1 , . . . , tr ) be a basis for H 0 (Pr , O(1)), and let s¯ = (s0 , s1 ) be a basis for H 0 (P1 , O(1)). A (t¯, s¯) - rigid stable family of bidegree (d,1) maps from 0 pointed, genus 0 curves to Pr × P1 consists of the data (π : C → S, {pi,j }0≤i≤r,1≤j≤d, {qi }0≤i≤1 , µ : C → Pr × P1 ) where: 1. (π : C → S, µ : C → Pr × P1 ) is a stable family of degree (d, 1) maps from 0 ponited, genus 0 curves to Pr × P1 (Definition 2.1.1). 2. (π : C → S, {pi,j }, {qi }) is a flat, projective family of d(r + 1) + 2 pointed stable curves in the sense of Deligne and Mumford.

65

3. For 0 ≤ i ≤ r, there is an equality of Cartier divisors µ∗ (ti ) = pi,1 + pi,2 + · · · + pi,d 4. For 0 ≤ j ≤ 1, there is an equality of Cartier divisors µ∗ (si ) = qi We then have the following proposition, analogous to Proposition 3 in [5]. The proof is almost identical, but is included because we refer to some of the steps later. Proposition 4.1.1. There is a fine moduli space M 0,0 (Pr × P1 , (d, 1), (t¯, s¯)) for the obvious functor M0,0 (Pr × P1 , (d, 1), (t¯, s¯)) of (t¯, s¯) - rigid stable maps. Moreover, this moduli space is a torus bundle over balanced subset of P1 [d(r + 1)]. Proof. Let m = d(r + 1) + 2. Let M 0,m be the Mumford-Knudsen compactification of the moduli space of m-pointed, genus 0 curves. Let π : U 0,m = M 0,m+1 → M 0,m be the universal curve with sections {pi,j } and {qi }. One exists since M 0,m is a fine moduli space. We have the the following r + 3 line bundles on U 0,m . For 0 ≤ i ≤ r, we have Hi = OU 0,m (pi,1 + pi,2 + · · · + pi,d ). For 0 ≤ j ≤ 1, we have Lj = OU 0,m (qj ) Let fi ∈ H 0 (U 0,m , Hi ) be the section representing the Cartier divisor pi,1 + pi,2 + · · · + pi,d , and let gj ∈ H 0 (U 0,m , Lj ) represent the Cartier divisor qi . 66

For any morphism γ : X → M 0,m , we form the fiber product: γ ¯

X ×M 0,m U 0,m −−−→ U 0,m     πX y πy γ

−−−→ M 0,m

X

We call the map (H, L)-balanced if 1. For 1 ≤ i ≤ r, πX ∗ γ¯ ∗ (Hi ⊗ H0−1 ) is locally free. 2. For 1 ≤ i ≤ r, the canonical map ∗ πX πX ∗ γ¯ ∗ (Hi ⊗ H0−1 ) → γ¯ ∗ (Hi ⊗ H0−1 )

is an isomorphism. 3. πX ∗ γ¯ ∗ (L1 ⊗ L−1 0 ) is locally free. 4. The canonical map ∗ ¯ ∗ (L1 ⊗ L−1 πX πX ∗ γ¯ ∗ (L1 ⊗ L−1 0 ) → γ 0 )

is an isomorphism. This means that the corresponding family of m pointed curves over X satisfies that every component of a curve has the same number of pi,j ’s on for every i, and has the same number of qj ’s on it. There is a Zariski open subscheme B ⊂ M 0,m satisfying 1. The inclusion of ι : B ֒→ M 0,m is (H, L)-balanced. 2. Every (H, L)-balanced morphism γ : X → M 0,m will factor uniquely through B. 67

Form the fiber diagram ¯ ι

B ×M 0,m U 0,m −−−→ U 0,m     πB y πy ι

−−−→ M 0,m

B

Let Gi = πB ∗¯ι∗ (Hi ⊗ H0−1 ), and let τi : Yi → B be the total space of the canonical C∗ bundle associated to Gi . Since Gi is locally free, it is the sheaf of sections of some line bundle. We just remove the zero section from Gi . We ∗ similarly define K = πB ∗¯ι∗ (L1 ⊗ L−1 0 ), and let ν : Z → B be the canonical C

bundle associated to K. The pull backs τi∗ (Gi ) and ν ∗ (K) have non-vanishing sections given by the fi and gi , and hence are trivial. Consider the product Y = Z ×B Y1 ×B Y2 ×B · · · ×B Yr . By identical arguments to [5], we see that Y represents the functor M0,0 (Pr × P1 , (d, 1), (t¯, s¯)). It’s clear that Y is a (C∗ )r bundle over Z by construction. We need only show that Z ⊂ P[d(r + 1)]. Form the fiber diagram ν¯

U −−−→   πZ y

U 0,m   πy

ν

Z −−−→ B ⊂ M 0,m .

The line bundles ν¯∗ (Li ) for 0 ≤ i ≤ 1 are canonically isomorphic to each other on U because ∗ ∗ ∗ −1 ν¯∗ (L1 ⊗ L−1 0 ) = πZ ν ι π∗ (L1 ⊗ L0 )

= (πZ∗ ν ∗ ι∗ )(i∗ πB ∗¯ι∗ (L1 ⊗ L−1 0 )) = πZ∗ ν ∗ πB ∗ πB ∗¯ι∗ (L1 ⊗ L−1 0 ) = πZ∗ ν ∗ (K) 68

and this is trivial by the note above. Therefore ν¯∗ (L0 ) = ν¯∗ (L1 ). By pull-back and these isomorphisms, we have two linearly independant sections ν¯∗ (gi ) of ν¯∗ (L0 ). This gives us a map µ : U → P1 . The sections {pi,j } and {qi } pull back to sections of U. Following [5], we see that (π : U → Z, {pi,j }, {qi }, µ : U → P1 ) is a universal d(r + 1) pointed stable family of curves with a map to P1 . So we have that Z ⊂ M 0,d(r+1) (P1 , 1). But, we know that M 0,d(r+1) (P1 , 1) = P1 [d(r + 1)] is the Fulton MacPherson compactification of d(r + 1) points on P1 by Example 2.1.6. As in [25] we see that for different values of s¯, we have different torus bundles over different open sets in M 0,d(r+1) (P1 , 1). The torus bundles can be glued together to obtain a torus bundle over a larger balanced open set that doesn’t depend on s¯ anymore. We will call this torus bundle G(t¯). We organize this information into the following diagram. (C∗ )r

(Sd )r+1

M 0,d(r+1) (P1 , 1) ⊃ B ←−−− G(t¯) −−−−→ Vt¯ ⊂ M 0,0 (Pr × P1 , (d, 1)) 4.1.2

Constructing Prd

We now continue by describing a t¯ cover of the linear sigma model Prd . This was studied by Mustat¸ˇa in his Thesis [25]. He gives the following definitions. Definition/Theorem 4.1.2. A d-acceptable family of morphism over S from P1 to Pr is defined by a scheme S, together with a line bundle L on S × P1 , and a morphism of sheaves e : Or+1 S×P1 → L 69

such that L|P1s = OP1s (d) for any s ∈ S and such that πS∗ e is nowhere zero, where πS is projection to S. Prd is a fine moduli space for the obvious functor of isomorphism classes of d-acceptable morphisms. This is a family of rational maps that is not undefined on any fiber. As before, the proof proceed by forming a rigid version of Prd , i.e. picking an ordering of the zeros. Choose a basis t¯ ∈ H 0 (Pr , O(1)). Definition/Theorem 4.1.3. A (t¯, d)- acceptable family of morphisms from P1 to Pr consists of a scheme S together with: 1. a line bundle L on S × P1 2. a morphism of sheaves e : Or+1 S×P1 → L with πS∗ e is nowhere zero, and 3. a set of sections {qi,j }0≤i≤r,1≤j≤d of S × P1 over S, such that via the natural isomorphism H 0 (Pr , OPr (1)) ∼ = H 0 (S × P1 , Or+1 S×P1 ), we have (e(t¯i ) = 0) =

d X

qi,j .

j=1

The obvious functor of isomorphism classes of (t¯, d) acceptable morphisms is represented by a (C∗ )r bundle over (P1 )d(r+1) we call Prd (t¯). We have the following diagram over the respective open sets. ∗ r

r+1

(C∗ )r

(Sd )r+1

(C ) (Sd ) P[d(r + 1)] ←−−− G(t¯) −−−−→ M 0,0 (Pr × P1 , (d, 1))       ¯ ϕ ϕ(t)y F −M y y

(P1 )d(r+1) ←−−− Prd (t¯) −−−−→ 70

Prd

where the vertical map on the left is the Fulton-MacPherson map, and the map in the middle is induced by the Givental map on the right. There is a natural action of G on the center spaces, pulled back from the action on the right. We take the G quotient of the above spaces in order to fill out Figure 4.1.

4.2

Relating the Rigid Covers

Using the above construction for Prd (t¯), we will be able to carry out a similar GIT construction for the rigid cover M (t¯) as was done for the global space M 0,0 (Pr , d) in Theorem 2.3.20. G(t¯) will play the role of M 0,0 (Pr × P1 , (d, 1)). We will show that M (t¯) = G(t¯)/G. First note that the Givental map ϕ : M 0,0 (Pr × P1 , (d, 1)) → Prd from Theorem 2.2.2 pulls back, giving a projective morphism we call ϕ(t¯) : G(t¯) → Prd (t¯). There is also an obvious forgetful map f (t¯) : G(t¯) → M (t¯) analagous to the forgetful map from Section 2.2. 4.2.1

The Group Action

Here we make explicit the actions of G on G(t¯), M (t¯) and Prd (t¯). The actions are just pulled back from the actions described on their global spaces in Section 2.3.1. 1. We start with Prd (t¯). The action is given by G × Prd (t¯) → Prd (t¯) g · (µ : P1 × S → Pr , {qi,j }) = (µ ◦ g −1 : P1 × S → Pr , {g(qi,j )}) 71

which is just the action from Prd pulled back from (Sd )r+1 , i.e. the quotient map p : Prd (t¯) → Wt¯ ⊂ Prd is G-equivariant. 2. The action on G(t¯) is defined similarly to the action on M 0,0 (Pr × P1 , (d, 1)). G × G(t¯) → G(t¯) g · (π : C → S, {pi,j }0≤i≤r,1≤j≤d, µ : C → Pr × P1 ) = (π : C → S, {pi,j }, (g × id) ◦ µ : C → Pr × P1 ) 3. The action on M(t¯) is then trivial, since we have no map to P1 . Since the actions are pulled back from the actions on the underlying moduli spaces (without the t¯), we have that ϕ(t¯) and f (t¯) are G- equivariant. We have the following diagram of G-equivariant morphisms.

G(t¯)   ϕ(t¯)y

f (t¯)

−−−→ M (t¯)

r+1

/(Sd ) Prd (t¯) −−−−−→

4.2.2

Prd

Taking the Quotient

As before, the main step in taking the quotient will be determining the stable locus of the G action on Prd (t¯). We will then use Theorem 2.3.18 of Hu to pull back the stable locus to G(t¯). Rather than doing the calculation on Prd (t¯), we pull back the stable locus from Prd by the following Theorem. It is analgous to Theorem 2.3.4 of [22] above, except we remove the requirement that X be proper. This Theorem was noted in [22] with no proof. 72

Theorem 4.2.1. Let f : X → Y be a geometric quotient by a finite group F , and assume that f is equivariant with respect to some reductive algebraic group G. Then, if L is a G-linearlized ample line bundle, we have X ss (f ∗ L) = f −1 {Y ss (L)}. Moreover, if Y s (L) = Y ss (L), we have X ss (f ∗ L) = X s (f ∗ L) = f −1 {Y ss (L)}. Proof. First recall that geometric quotients are preserved under flat pull back (Remark 7 in Chapter 0 of [22]). By this we mean that if f : X → Y is a geometric quotient, and π : Y ′ → Y is a flat map, then the induced map f¯ : X ×Y Y ′ → L is a geometric quotient. f¯

X ′ = X ×Y Y ′ −−−→ Y ′     πy π ¯y f

−−−→ Y

X

This means that the map on functions OL → f¯∗ OX ′ induces an isomorphism onto the subring of F -invariant functions. But we argue that sections of Y ′ are actually F invariant sections of X ′ . ˇ the dual of a line bundle on Y . The reason for the dual will be Take L apparent soon. For any affine open set of U ⊂ Y with OY (U) = A, the inverse ˇ is an affine open set with coordinate ring image in L OLˇ (π −1 (U)) = SymA Γ(U, L) [6]. However, this is equal to the Rees algbebra. SymA Γ(U, L) = A ⊕ Γ(U, L) ⊕ Γ(U, L)2 ⊕ . . . 73

ˇ →L ˇ is also a geometric quotient, we know that Now, since the map f¯ : f ∗ L there is an isomorphism on global functions ˇ = (SymO Γ(X, f ∗ L)) ˇ F SymOY Γ(Y, L) X that preserves the grading. Looking at the linear pieces, we see that Γ(Y, L) = Γ(X, f ∗ L)F . We now prove the theorem. Note that we automatically have X ss (f ∗ L) ⊇ f −1 {Y ss (L)}. Moreover, we can assume that Y = PN and L = O(1). We know by Proposition 1.7 in [22] that there is an embedding I : Y → PN , along with a G action on PN and a G linearization on O(1) such that I is G invariant and Ln = I ∗ O(1). For, assume that we’ve proven the theorem for I ◦ f : X → PN . Then we have X ss (f ∗ L) = X ss ((I ◦ f )∗ O(1)) = (I ◦ f )−1 {(PN )ss (O(1))} = f −1 ◦ I −1 {(PN )ss (O(1))} ⊆ f −1 {Y ss (L)} Take x ∈ X ss (f ∗ L). We will show that f (x) is semi-stable in Y . Recall that this means there exists m > 0 and s ∈ Γ(Y, Lm )G such that Ys = {y ∈ Y : s(y) 6= 0} is affine and contains f (x). F acts on X and so we look at the orbit of x under this action. Call this orbit O F (x). We know by Theorem 2.3.4 that there is a quasi-projective geometric quotient p : X ss (f ∗ O(1)) → X ss (f ∗ O(1))/G, and we examine the image of the 74

orbit, p(O F (x)). This is a finite set of points in a quasi-projective variety, and so we can find a hyperplane H that misses all the points. This hyperplane pulls back to a section h ∈ Γ(X, f ∗ L) that is G equivariant automatically. Moreover, h(xi ) 6= 0∀xi ∈ O F (x). We construct a section that is F equivariant, by averaging over the group F . Specifically we define hX =

X

σh.

σ∈F

Then hX is still G equivariant, and now is also F equivariant, and doesn’t vanish at any of the points of O F (x). However, by the work that we’ve done, we see that F invariant sections are the same as sections of L. hPN ∈ Γ(PN , O(1))G such that hPN (f (x)) 6= 0. And we have that (PN )hPN is affine since it’s the complement of a hyperplane. This gives us that X ss (f ∗ L) = f −1 {Y ss (L)} Now, if Y ss (L) = Y s (L), we need only notice that X s (f ∗ L) ⊇ f −1 {Y s (L)} by Proposition 1.18 in [22], and X s (f ∗ L) ⊆ X ss (f ∗ L) to conclude that X s (f ∗ L) ⊂ X ss (f ∗ L) = f −1 {Y s (L)}.

Thus, when we have a finite group quotient, the stable (and semi-stable) loci exactly pull back. We apply this Theorem to our situation of the (Sd )r+1 quotient q : Prd (t¯) → Wt¯ ⊂ Prd , where the map is the quotient by (Sd )r+1 . 75

Corollary 4.2.2. (Prd (t¯))ss (q ∗ O(1)) = q −1 {(Prd )ss (O(1))} The closed points correspond to r + 1 forms of degree d in two variables, and an ordering of the zeros of those forms such that the r +1 don’t have a common zero of multiplicity > d/2. We now state our rigid version of Theorem 2.3.20. Proposition 4.2.3. Let M be a relatively ample linearized line bundle on G(t¯), relative to q ∗ O(1). Then there exists a n0 such that L = (q ∗ O(n)) ⊗ M) is ample for all n ≥ n0 . Moreover (G(t¯))s (L) = (G(t¯))ss (L) = ϕ(t¯)−1 {(Prd )ss (q ∗ O(1)}. In addition we have (G(t¯))ss (L) = M (t¯) G and a commutative diagram (G(t¯))ss (L)   ¯ ϕ(t)y

f (t¯)

−−−→

M (t¯)   y

(Prd (t¯))ss (q ∗ O(1)) −−−→ (Prd (t¯))ss (q ∗ O(1))/G where f (t¯) and ϕ(t¯) were defined above. Proof. The first part of the Theorem again follows by an application of Hu’s Theorem 2.3.18. The rest will follow by similar arguments to Theorem 2.3.20, however, we can’t speak of Picard numbers because the spaces are all open.

76

However, since we’re just looking at one linearization of one line bundle (instead of many as in 2.3.20) we can just check things by hand. Following the notation from [15], let U be the semi-stable locus in Prd (t¯) for the above action of G. Recall that this corresponds to r + 1 degree d homogenous polynomials in two variables with no common root of multiplicity > d/2 along with an ordering of the zeros. Let U ′ = ϕ(t¯)−1 (U). Let the corresponding quotients be Q and Q′ . We have the obvious composition of G invariant maps: U ′ → U → Q. By the universal properties of GIT quotients, we get a proper birational map Q′ → Q. Similarly, since G acts trivially on M 0,0 (Pr , d, ¯t), we have by the universal property again a proper birational map from Q′ → M 0,0 (Pr , d). We will show that this is an isomorphism by showing that no divisor is collapsed. We look at the inverse image of points of M (t¯) in G(t¯) and they consist of only one orbit. Just as before, there are many components in G(t¯) that map to any component of M (t¯). From Proposition 2.3.3, there are 3k − 1 components of G(t¯) mapping to a t¯-rigid stable map with domain curve with k components. However, only one of these 3k − 1 components will lie in the stable locus, by the following Lemma. Lemma 4.2.4. Let C be a connected, genus-0 curve such that each edge is P labeled with a degree di . If d = di is odd, then there is a unique irreduciable ¯ no tooth has sum of component C¯ such that if C is a comb with handle C, degrees ≥ d/2.

77

Proof. Refer to Definition 2.2.1 for the definition of a comb. Let {C}d/2 be the set of all connected subcurves of C with degree ≥ d/2. Intersect all such subcurves. Note that the intersection is non-empty. Otherwise, there would be two disconnected subcurves of degree ≥ d/2. Since d is odd, the sum would thus be > d, a contradiction. We now argue that the intersection is unique. Assume that the intersection consists of two irreduciable curves E1 , E2 of degrees d1 , d2 . They must be connected, else our starting curve wasn’t of genus 0. So they’re adjacent, and we can form the following picture, with degrees as labeled. First see that A

B

d2

d1

dB

dA

E

E1

2

Figure 4.2: C¯ ⊂ C dA + d1 ≤ d/2, else A ∪ E1 is a subgraph of degree ≥ d/2, not containing E2 , which is a contradiction. Similarly, dB + d2 ≤ d/2. Then dA + dB + d1 + d2 < d a contradiction. This unique intersection is C¯ since no tooth meeting it can have degree ≥ d/2. So, given any map in M (t¯), there is a unique component in the inverse image that intersects the stable locus in G(t¯). This component is clearly a 78

single orbit since the degree one part in G(t¯) can be any element of G. Remark 4.2.1. We think of a closed point in Prd as [f0 , . . . , fr ], and a point of Prd (t¯) as the map along with an ordering of the “zeros”. The map to (P1 )d(r+1) sends the ordered zeros of f0 to the first d points of (P1 )d(r+1) , the ordered zeros of f1 to the second d points of (P1 )d(r+1) , etc. When we pull back the stable locus from Prd to Prd (t¯) we see that the corresponding stable set corresponds to r + 1 degree d forms, with no common root of degree ≥ d/2, along with an ordering of those zeros. We can then push this stable locus forward to (P1 )d(r+1) . The image of the stable locus will be the open set where the obvious (r + 1) groups do not have a common point of multiplicity ≥ d. Notice that this doesn’t say much about the total multiplicity that a point can appear with in (P1 )d(r+1) . For example the first r sets of d points can be identical, as long as the last set of d points doesn’t have more than d/2 points in common, i.e. a point can appear with multiplicity as large as dr + (d − 1)/2. Now recall from [15] and [22] that the semi-stable locus for the usual diagonal action of G on (P1 )d(r+1) consists of d(r + 1) ordered points, none repeated more than d(r + 1)/2 times. We see that the image of the pull back of (Prd )s (moving from the right to the left of the diagram) contains this set. Thus, we don’t get a map from Prd (t¯)s → (P1 )d(r+1) /G. Similarly, we have a map from G(t¯) → B ⊂ P1 [d(r + 1)]. However, again if we push forward the stable set to P1 [d(r + 1)], we will land outside the stable locus of the diagonal action pulled back from (P1 )d(r+1) . 79

Using Proposition 4.2.3, the construction of G(t¯), and the above remark we see that we have filled Figure 4.1 to the following (over restricted open sets). P1 [d(r + 1)] .................................................................................................. G(t¯) ........................................................................ M 0,0 (Pr × P1 , (d, 1)) ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ......... ..

F-M

M 0,d(r+1)

... .... .... ... .... ... ... ... .... ... .... ... ... .... ... .... ... ... ... .... ... .... ..... ... ...... ... ... ... . . . . ...................................................................................................... ... ... ... ... ... ... ... ... ... ... . ... .......... .

M (t¯)

... ... .... ... ... ... .... ... .... ... ... .... ... .... ... ... ... .... ... .... ... ... ..... ... ........ ... r+1 . . d .... . . . . .............................................................................................. ... 0,0 .... ... ... ... ... ... ... ... ... . ... ......... .

(S )

... ... ... ... ... ... ... ... ... ... ... ... ... . ............................................................................................................................... ... .... .... ... ... ... .... .... ... .... ... .... ... .... ... .... ... .... .... ..... .... ..... .. ......... ........ .

(P1 )d(r+1) ........................................................................................................ Prd (t¯)

Prd (t¯) G

ϕ

M

Prd

(Pr , d) ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ........ .

ϕ¯

... .... .... .... .... .... .... .... .... .... .... ...... ......

................................................................................................................................

Prd G

Figure 4.3: The GIT Quotients

4.3

Factoring M 0,0(Pr , d)

Our goal initially was to construct M 0,0 (Pr , d) as sequence of blow ups of some projective variety, analogous to M 0,n in [16], or X[n] in [7] . We immediately saw that this wouldn’t work because M 0,0 (Pr , d) is singular. However, it is smooth when considered as a stack ([5]), and the map from 2.3.25 is exactly the map we try to factor into a stack analogue of smooth blow-ups. We start by looking at M 0,0 (Pr × P1 , (d, 1)) and the Givental map to Prd and attempt to factor this map. Again, this won’t work because M 0,0 (Pr × P1 , (d, 1)) is still singular, and Prd is smooth. So we look at the t¯ rigidifications constructed above, where the spaces are smooth. We will use the blow up construction of the Fulton-MacPherson as motivation [7]. 80

4.3.1

Factoring P1 [(r + 1)d] → (P1 )(r+1)d

Recall from [7] that in general we can construct X[n] as a sequence of 2n −n−1 blow-ups of X n . Mustat¸ˇa makes a simplification in the case when X = P1 . We start with the following definitions. Here, let N = d(r + 1). Definition/Theorem 4.3.1. [25] We call a degree one morphism φ from a rational curve C with N marked points to P1 k-stable if 1. no more than N − k points coincide 2. any ending irreduciable component in C, except the parametrized one contains more than N − k marked points ( by an ending component, we mean a component such that if we remove it, C will remain connected). 3. all marked points are smooth, and any component of C contracted by φ has at least 3 distinct special points on it (marked points or nodes). There is a smooth projective moduli space P1 [N, k] for families of k-stable degree 1 morphisms. Moreover, P1 [N, k] is the blow-up of P1 [N, k − 1] along the strict transforms of the k-dimensional diagonals in (P1 )N . Remark 4.3.1. We can define the obvious moduli functor P1 [N, k]. Notice that P1 [N, 0] (0-stable maps) is represented by (P1 )N and that P1 [N, N − 1] ((N − 1)-stable maps) is represented by the Fulton-MacPherson space P1 [N]. This allows a nice geometric description of the blow ups necessary to construct P1 [N]. For example, P1 [N, 1] is the blow up of P1 [N, 0] = (P1 )N along the one dimensional large diagonal ∆ ⊂ (P1 )N . This corresponds to a parametrized P1 81

with N points all coinciding. The inverse image of ∆ in P1 [N, 1] corresponds to a curve with two components where one of the components is parametrized, and the other contains all N points, configured in such as way that not all N points coincide.

All points coincide

Figure 4.4: The First Blow Down Remark 4.3.2. While all the blow ups P1 [N, k] → P1 [N, k − 1] are necessary to construct P1 [N] from (P1 )N , we won’t be concerned with all of them. This is because G(t¯) is torus bundle only over the balanced subset of P1 [N] corresponding to (π : C → S, {pi,j }0≤i≤r,1≤j≤d , µ : C → P1 |#(pi,j ) = #(pi′ ,j ) on every component of C). This means that (amongst other things) every component of C will have a multiple of (r + 1) points on it. As such, we need not blow up at stages that would give us configurations without multiples of (r + 1) points on components. Of course, even this is too much, because the balanced subset doesn’t just require a certain number of points, but the way that these points are labeled is important as well.

82

4.3.2

Factoring ϕ(t¯) : G(t¯) → Prd (t¯)

Here we pull back the above factorization to the map ϕ(t¯) : G(t¯) → Prd (t¯). We start this by forming the fiber product over the balanced subset from Remark 4.2.1. P1 [N, N − (d − 1)(r + 1)] ←−−− P1 [N, N − (d − 1)(r + 1)] ×(P1 )N Prd (t¯)     blow up y y (P1 )N ⊃ B

(C∗ )r

Prd (t¯)

←−−−

We call this blow up Prd (t¯, d − 1). This is a torus-bundle over P1 [N, N − (d − 1)(r + 1)]. We then inductively form the fiber diagrams to get the following tower (C∗ )r

P1 [N, N − 1]   y

←−−−

(C∗ )r

P1 [N, N − (r + 1)]   y .. .   y

←−−−

P1 [N, 0]

←−−−

(C∗ )r

←−−−

∗ r

G(r, d, ¯t)   y Prd (t¯, 1)   y .. .   y

(C ) P1 [N, N − (d − 1)(r + 1)] ←−−− Prd (t¯, (d − 1))     y y (C∗ )r

Prd (t¯)

It’s clear that these intermediate spaces are moduli spaces. In fact, we have that Prd (t¯, n) is a fine moduli space for maps of the following form. 83

Definition/Theorem 4.3.2. A (t¯, d, k) - acceptable family of morphism over S is given by the following data (π : C → S, φ : C → P1 , {qi,j }0≤i≤r,1≤j≤d , L, e) where 1. The family (π : C → S, φ : C → P1 , {qi,j }0≤i≤r,1≤j≤d ) is a (r+1)(k−1)+1 stable family of degree 1 morphisms to P1 . 2. L is a line bundle on C. 3. e : Or+1 → L is a morphism of sheaves with π∗ e nowhere zero and that, C via the natural isomorphism H 0 (Pn , O(1)) ∼ = H 0 (S × P1 , Or+1 S×P1 ) we have (e(t¯i ) = 0) =

d X

qi,j

j=1

There is a smooth moduli space Prd (t¯, k) for these families that is a torus bundle over an open subset of P1 [(r + 1)d, (r + 1)(k − 1) + 1]. Thus, immediately we get the following factorization of ϕ(t¯). Then in identically the same way as [7], Mustat¸ˇa glues together the Prd (t¯, k) for different values of t¯ in order to obtain global objects that factor ϕ. Definition/Theorem 4.3.3. A (d, k) - acceptable family of morphism is given by the following data (π : C → S, µ = (µ1 , µ2 ) : C → Pr × P1 , L, e) where: 84

P1 [N] = P1 [N, N − 1] .................................................................. G(t¯) ........................................................................ M 0,0 (Pr × P1 , (d, 1)) ... . ......... ..

... . ......... .

... . ......... ..

... . ......... ..

.... ... .... ... .... .... ... .... .... ... .... .... ... . .........

P1 [N, (r + 1)(d − 2) + 1] ................................... Prd (t¯, d − 1) .. ....

.. ....

. ......... ..

P1 [N, 1] ... . ......... ..

. .......... .

.............................................................................................

M (t¯)

... ... ... ... ... ... ... ... ... ... ... ... ... . ................................................................................................................................. ... .... .... ... ... .... ... ... ... .... ... .... ... ... .... ... .... ... ... .... ..... .... ... . ...... . . . . . . . .... ..

Prd (t¯, 1) ... . ......... ..

... ... .... ... .... ... ... ... .... ... .... ... ... .... ... .... ... ... ... .... .... ... ..... ... ....... ... ... ... . . . . . . ........................................................................................... ... 0,0 .... ... ... ... ... ... ... ... ... ... . ......... .

(P1 )N = P1 [N, 0] ................................................................................. Prd (t¯)

ϕ

M

Prd

(Pr , d) ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ......... .

ϕ¯

... .... .... ... .... .... ... .... .... ... .... .... ..... ......

Prd (t¯)/G .......................................................................................................... Prd /G

Figure 4.5: Factoring ϕ(t¯) 1. L is a line bundle on C which, together with the morphism e : Or+1 →L C determines the rational map µ1 : C 99K Pr . 2. For any s ∈ S and any irreducible component C ′ of Cs , the restriction eC ′ : Or+1 C ′ → LC ′ is non-zero. 3. For any s ∈ S, degLCs = d and the image eCs (H) ∈ H 0 (Cs , LCs ) of a generic section H ∈ H 0 (Cs , Or+1 Cs ) determines the structure of a (r + 1)(k − 1) + 1 stable morphism on µ2 : Cs → P1 There is a projective coarse moduli space Prd (k) for these objects. It’s these Prd (k) that we will take the G quotient of. The next section will push these intermediate moduli spaces forward to factor ϕ¯ : M 0,0 (Pr , d) → Prd /G. 85

P1 [N] = P1 (N, N − 1) ................................................................ G(t¯) ........................................................................ M 0,0 (Pr × P1 , (d, 1)) ... . ......... ..

... . ......... .

... . ......... ..

... . ......... ..

. ......... ..

. .......... .

.... .... .... .... ... .... ...................................................................................... .... ... .... .... ... .... . ..........

P1 [N, (r + 1)(d − 2) + 1)] ................................ Prd (t¯, d − 1) .. ....

.. ....

P1 [N, 1]

.............................................................................................

... . ......... ..

... . ........ .

... .... .... ... .... .... ... .... .... ... .... .... ..... .......

Prd (d − 1) ... . ........ .

M (t¯) .................................................................................................... M 0,0 (Pr , d)

... ... ... ... ... .............................................................................................................. ... ... ... ... . ..... .. ................................................................................................................................. ... ... .... ... ... ... .... ... .... .... ..... .... .... .... .... ... .... ... .... ... .... .. ..... ........ ........ .

... . ......... .

Prd (t¯, 1)

Prd (1)

... . ......... ..

... . ........ .

(P1 )N = P1 (N, 0) .............................................................................. Prd (t¯)

Prd (t¯) G

Prd

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ......... .

ϕ¯

... .... .... ... .... .... ... .... .... ... .... .... ..... ......

.......................................................................................................................

Prd /G

Figure 4.6: Factoring ϕ 4.3.3

Factoring ϕ¯ : M 0,0 (Pr , d) → Prd /G

In order to factor the map M 0,0 (Pr , d) → Prd /G we could proceed in two different directions. 1. On one hand, we could take the quotient of the Prd (t¯, k) by G. This will give us intermediate moduli spaces factoring ϕ( ¯ t¯). After we obtain these spaces, we can glue together for different values of t¯, obtaining global spaces factoring ϕ. ¯ 2. On the other hand we can take the global spaces Prd (k) that were already constructed by [23] above. Taking the quotient of these spaces by G, will factor ϕ. ¯ Both of these constructions will give the same space because quotient by the finite group (Sd )r+1 commutes with the action of G. We define the

86

objects that we classify, and show that the obvious quotient is a coarse moduli space for these objects. Definition 4.3.1. A (d, k)∗ acceptable morphism is given by the following data. (π : C → S, µ : C → Pr , L, e) where: 1. L is a line bundle on C which, together with the morphism e : Or+1 →L C determines the rational map µ : C → Pr . 2. For any s ∈ S and any irreducible component C ′ of Cs , the restriction eC ′ : Or+1 C ′ → LC ′ is non-zero. 3. For any s ∈ S, degLCs = d and the image eCs (H) ∈ H 0 (Cs , LCs ) of a generic section H ∈ H 0 (Cs , Or+1 Cs ) determines the structure of a (r+1)(k− 1)+1 - stable rigid morphism where C¯s plays the role of the parametrized component in the definition of a n stable degree 1 morphism above. Recall that C¯s is the unique component of Cs such that no tooth has degree > d/2. (Lemma 4.2.4). As usual, we can define the moduli functor corresponding to these families. Namely we can define the functor M0,0 (Pr , d, k) from {Schemes} to {Sets}, which takes a scheme S to the set of isomorphism classes of (d, k)∗ acceptable morphisms over S. We then have 87

Theorem 4.3.4. There is a projective coarse moduli space M 0,0 (Pr , d, k) for families of (d, k)∗ acceptable morphisms. Proof. We show that M 0,0 (Pr , d, k) := (Prd (k))s /G satisfies the properties of a coarse moduli space. This quotient is constructed identically to that from Theorem 2.3.20. We pull back the stable locus from Prd by 2.3.18. Again, the stable locus in Prd (k) will be those (d, k) - acceptable maps such that no tooth has degree > d/2. The universal properties of this quotient space are inherited from the universal properties of Prd (k) as well as the universal properties of a categorical quotient. That the closed points correspond 1 − 1 to (d, k)∗ stable maps is obvious. First, we need to construct a natural transformation of functors φ : M0,0 (Pr , d, k) → HomSch (∗, M 0,0 (Pr , d, k)) where M0,0 (Pr , d, k) is the functor {Schemes} → {Sets} above. Given a family of (d, k)∗ - acceptable morphisms (π : C → S, µ : C → Pr , L, e) we can get a (d, k) - acceptable morphism (π : C → S, µ = (µ1 , µ2 ) : C → Pr × P1 , L, e) by taking µ2 : C → P1 to be identity on C¯s and constant on the other components. This will lie in the stable locus by construction, and thus gives 88

a map S → (Prd (k))s . Composing with the quotient gives an element of HomSch (S, M 0,0 (Pr , d, k)). Now, we need to show that if given a scheme Z and a natural transformation of functors ψ : M0,0 (Pr , d, k) → HomSch (∗, Z), there exists a unique morphism of schemes γ : M 0,0 (Pr , d, k) → Z such that ψ = γ˜ ◦ φ. So assume we are given Z and ψ. By the above, we have a transformation M0,0 (Pr , d, k) → (Prd (k))s . As such we get a transformation ψ¯ : (Prd (k))s → HomSch (∗, Z) which by representability gives a map γ¯ : (Prd (k))s → Z. This map is G equivariant by construction, hence factors though the quotient γ : M 0,0 (Pr , d, k) → Z

Corollary 4.3.5. M 0,0 (Pr , d, k) is the weighted blow up along a regular local embedding of M 0,0 (Pr , d, k − 1). We can sum up this construction with the following figure

89

(C∗ )r (Sd )r+1 P1 [N] = P1 (N, N − 1) ................................................................ G(t¯) ........................................................................ M 0,0 (Pr × P1 , (d, 1)) ... . ......... ..

... . ......... .

... . ........ .

... . ......... ..

... . ......... ..

... . ........ .

... .... .... ... .... .... ... .... .... ... .... ... .... .... ..... .... ....... ... .... .... ... .... .... ... 0,0 .... .... ..... ......

P1 [N, (r + 1)(d − 2) + 1)] ................................ Prd (t¯, d − 1) ..................................................................................... Prd (d − 1) .. ....

.. ....

... . ......... ..

. ........ .

. ......... .

. ......... ..

P1 [N, 1]

.. ....

.............................................................................................

(C∗ )r

M

(Pr , d) ... . ......... .

Prd (t¯, 1) ............................................................................................................ Prd (1) M 0,0 (Pr , d, d − 1) ... . ......... ..

(Sd )r+1

... . ........ .

(P1 )N = P1 (N, 0) .............................................................................. Prd (t¯) ............................................................................................................................. Prd

... .... .... ... .... .... ... .... .... ... .... ... .... .... .... .... ... ....... .... .... ... .... .... 0,0 ... .... .... ..... .......

M

... . ......... .

.. ....

. .......... .

(Pr , d, 1) ... . ......... .

Prd /G Figure 4.7: Factoring ϕ¯ Remark 4.3.3. Notice that Prd /G = M 0,0 (Pr , d, 1) = · · · = M 0,0 (Pr , d, d+1 ). 2 This is because up to that point, the exceptional loci of the blow ups will lie outside the stable locus. For example, the exceptional divisor of Prd (t¯, 1) → Prd (t¯) corresponds to a curve with two components. One is parametrized, and the other has all the d(r + 1) points on it. This clearly lies outside the stable locus.

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Chapter 5 A Word About Stacks

We have seen that the correct language to speak about M 0,0 (Pr , d) is as a stack, and not as a scheme. The reason is that as a scheme, the space is singular. However, as a stack, the space is smooth. We would like to know that the same results hold in the language of stacks. Here we include the (lenghty) definition of a stack. It is mainly included as a reference for the reader. All information was taken from [10], [4], or [32].

5.1

Definitions

Fix a scheme S, and denote by S the category of S schemes. Let F be a category along with a fixed covariant functor pF : F → S, i.e. a category over the category of S -schemes. We say that an object X ∈ Ob(F) lies over B ∈ Ob(S) if pF(X) = B. We will write F(B) for the objects that lie over a scheme B. We use similar language for morphisms. Definition 5.1.1. [10] We say that pF : F → S is fibered into groupoids (or just a groupoid) if 1. For any {B ′ → B} ∈ Mor(S) and any object X ∈ F(B), there is an object X ′ ∈ F(B ′) and a morphism {X ′ → X} ∈ F lying over {B ′ → B}. 91

2. For any diagram of S morphisms in S B1 .......................................................................................................................................................B . 2 ....... ....... ....... ....... ......... .......

B3

.. ....... ....... ....... . ........ ..........

and any objects Xi ∈ F(Bi) in a compatible diagram in F X1 ...........

X2

..... ....... ........ ....... ....... . . . . . . . . .........

....... ....... ....... ........ . ......... .. .

X3 there is a unique morphism {X1 → X2 } lying over {B1 → B2 } making this diagram commute. Remark 5.1.1. Notice that if {B ′ → B} is an isomorphism in S, then any morphism in F lying over {B ′ → B} is also an isomorphism. So for a fixed scheme B, F(B) is a groupoid( i.e. a category where all morphisms are isomorphisms.) Remark 5.1.2. Condition (2) above asserts that the element X ′ from (1) is unique up to unique isomorphism. For every f : B ′ → B, and every X ∈ Ob(F(B), we choose once and for all such an X ′ , and call it f ∗ X. Moreover, condition (2) implies that if {s : X ′ → X} is a morphism in F(B), then there is a canonical morphism {f ∗ s : f ∗ X ′ → f ∗ X} in F(B ′ ). So, given a morphism f : B ′ → B there is an induced covariant functor f ∗ : F(B) → F(B ′ ). Remark 5.1.3. In the case of moduli problems, where the objects of F are flat families, and the morphisms are fiber diagrams, then F is automatically fibered into groupoids, where pF : F → S is projection onto the base of the family. Remark 5.1.4. Given any S-scheme B, the functor of points B is a groupoid. The objects are X-points {X → B}, and a morphism from {X → B} to 92

{X ′ → B} is a morphism {X → X ′ } that commutes with projection to B. The functor pB : B → S forgets the B structure and just considers them as S-schemes. S is trivially a groupoid, as the fiber over a scheme is a set, and hence a groupoid. There is a very important functor that is essential in the definition of a stack. Definition 5.1.2. Let pF : F → S be a groupoid. Let B ∈ Ob(S), and let X, Y be any objects in the fiber F(B). Define the contravariant functor IsoB (X, Y ) : B → {Sets} by associating to {f : B ′ → B} ∈ Ob(B), the set of isomorphisms in F(B ′ ) between f ∗ X and f ∗ Y . To see the action on morphisms, let {f : B ′ → B} and {g : B ′′ → B} be two objects of B. Let h : B ′′ → B ′ be a morphism in B between these points. Recall this means that g = f ◦ h. We would like to see what map of sets IsoB (X, Y )(B ′ ) → IsoB (X, Y )(B ′′ ) this gets sent to. Recall by (2) in 5.1.1, we know that there are unique isomorphisms ψX : g ∗X → h∗ f ∗ X

ψY : g ∗Y → h∗ f ∗ Y.

Take an isomorphism φ : f ∗ X → f ∗ Y ∈ IsoB (X, Y )(B ′ ). By Remark 5.1.2, we know that h∗ : F(B ′ ) → F(B ′′ ) is a covariant functor. As such, we can pull back and obtain an isomporphism h∗ φ : h∗ f ∗ X → h∗ f ∗ Y. Then the composition ψY−1 ◦h∗ φ◦ψX is an isomorphism between g ∗X and g ∗ Y , and is the image of φ in IsoB (X, Y )(B ′′ ). 93

The last step before we give the definition of a stack is to describe a topology on the category S that we will use. The usual Zariski topology is too coarse. The correct topology turns out to be a Grothendieck topology, either the ´etale topology, or the fppf topology. We only define the ´etale topology here as that will lead us to a Deligne-Mumford stack, while fppf would lead us to an Artin stack. Definition 5.1.3. An open covering in the ´etale topology consists of a collection of ´etale morphisms {bi : Bi → B} such that {∪Bi → B} is surjective. Form the fiber diagram bji

Bij = Bi ×B Bj −−−→ Bj     bj y bij y b

i −−− → B

Bi

Since being ´etale is invariant under base change, {bij : Bij → Bi } is an open cover for fixed i. We then have Definition 5.1.4. A stack over S is a category fibered into groupoids pF : F → S such that 1. (Isomorphisms are a sheaf) For all B ∈ S, and all X, Y ∈ F(B), IsomB (X, Y ) : B → {Sets} is a sheaf in the ´etale topology. In other words, for all B ∈ S, all X, Y ∈ F(B), all open covers {bi : Bi → B}, and all isomorphisms αi : b∗i X → b∗i Y such that b∗ij αi = b∗ji αj ∈ F(Bij ), there is a unique isomorphism α : X → Y such that b∗i α = αi . 94

2. (Decent data is effective) Let {bi : Bi → B} be an open cover, and Xi ∈ Ob(F(Bi)). Suppose that we have isomorphisms φij : b∗ij Xi → b∗ji Xj in F(Bij ) satisfying the cocycle condidion. Then there is X ∈ Ob(F(B)) such that b∗i X ∼ = Xi . Definition 5.1.5. A morphism of stacks f : F → G is a functor between the categories such that pG ◦ f = pF. If f is an equivalence of categories, then we say that F and G are isomorphic. The following very important Lemma shows why stacks are so important for moduli problems Lemma 5.1.1. [4] Let F be a stacks and B a scheme. The functor b : HomS (B, F) → F(B) which sends {f : B → F} ∈ HomS (B, F) to f (id : B → B) ∈ F(B) is an equivalence of categories This means that an object of F(B) is equivalent to giving a morphism of B into the stacks F i.e. the stack behaves like a moduli space. Up to now, everything has been strictly categorical. In order to discuss any geometric ideas, we need to place some restrictions on our stacks. This leads us to the definition of a Deligne-Mumford stack. We start with the notion of the fiber product of stacks.

95

Definition 5.1.6. [10] Given two stack morphisms f : F → G and H : H → G, we define a new stacks F ×G H (along with projections to F, H) as follows. Objects are triples {(X, Y, α) | X ∈ F(B), Y ∈ H(B), {α : f (X) → h(Y )} ∈ Mor(G(B))} A morphism between (X, Y, α) and {(X ′ , Y ′ , α′ ) | X ′ ∈ F(B ′), Y ′ ∈ H(B ′ ), {α′ : f (X ′ ) → h(Y ′ )} ∈ Mor(G(B ′ ))} is a pair of morphisms (a, b) with a : X → X′

b: Y →Y′

lying over the same morphism of schemes in S, and such that α′ ◦ f (a) = h(b) ◦ α′ . Remark 5.1.5. While it is true that the fiber product of stacks satisfies the universal property analgous to that of fiber products of schemes, it is not true that the obvious square

q

F ×G H −−−→ H     py hy f

F −−−→ G commutes, since f p(X, Y, α) = f (X) and hq(X, Y, α) = h(Y ), and these need not be equal in G, only isomorphic. However, there is a natural transformation of functors f p → hq, namely α. Remark 5.1.6. Recall above we showed that the functor of points B associated to an S-scheme B was a groupoid. In fact, it is a stack. That isomorphisms are a sheaf is clear, and in fact holds in the Zariski topology. That decent data 96

is effective is more complicated. The stack associated to a scheme is a stack with no automorphisms. Definition 5.1.7. A stack F is represented by a scheme B if F is isomorphic to the stack B. If F is represented by B, we say F has property “P” if B does. A morphism of stacks f : F → G is representable if for all objects B ∈ S and morphisms B → G, the fiber product B ×G F is representable by a scheme. We say that a morphism f has property a local property “P” if for every B → G, the pullback B ×G F → B has that property. We are now able to provide the definition of a Deligne-Mumford Stack Definition 5.1.8. A stack F is Deligne-Mumford if 1. The diagonal ∆F : F → F ×S F is representable, quasi-compact, and separated. 2. There exists a scheme U and an ´etale, surjective, morphism u : U → F There is a lot of information encoded in the diagonal morphism, as the following Lemma illustrates. We will only consider Deligne-Mumford stacks from now on. Lemma 5.1.2. Let F be a stack and let ∆F : F → F ×S F be the diagonal morphism. Let B ∈ S, and let B be the stack of Schemes over B. Let f : B → F → F ×S F be a morphism. By 5.1.1, we know this is equivalent to an object of F → F ×S F over B, which is just two elements X, Y ∈ F(B) since there are no automorphisms in S. Then F ×F×S F B ∼ = IsomB (X, Y ). 97

Proof. Note that since S is a scheme, and has no automorphisms except for identity, we have that F ×S F = F × F. Let U ∈ S. A object of F ×F×F B over U is a pair of objects with the condition [(a, b) | {a : U → B} ∈ B(U) , b ∈ F(U) , f (a) ∼ = ∆F(b) in F × F]. Now, f (a) = (a∗ X, a∗ Y ) ∈ F × F, and ∆(b) = (b, b) ∈ F × F. Saying that (a∗ X, a∗ Y ) ∼ = (b, b) in F × F means that there are two morphisms (isomorphisms automatically since they’re over the identity) ϕ1 : a∗ X → b

ϕ1 : a∗ Y → b

which gives us an isomorphism a∗ X → a∗ Y . The requirements on the diagonal from 5.1.8 can then given some geometric intuition from the following Corollary. It explains why we put the stability conditions on our degree d maps in 2.1.1 Corollary 5.1.3. [32] If F is a Deligne-Mumford stack, B quasi-compact, and X ∈ F(B), then X has only finitely many automorphisms.

5.2

Group Actions on Stacks

As mentioned in Chapter 2, the correct language for speaking of M 0,n (Pr , d) and M 0,n (Pr × P1 , (d, 1)) is stacks and not schemes, because in this category they are smooth, while as schemes they are singular. In addition, there are no universal curves over the schemes (as they are not fine moduli spaces). It is helpful to know that when we work in the language of stacks we obtain 98

the same answers as using GIT. In [29], group actions on stacks are studied. Namely we have. Definition 5.2.1. Let F be a stack over S, and G be a sheaf in groups over S. Let m be multiplication in G and e it’s unit section. Then an action of G on F is a morphism of stacks µ : G × F → F satisfying the 1 - diagrams G×G×F ... ... ... ... ... ... ... ... ... .. ........ .

m × idF

...........................................................................................

G×F µ

idG × µ

G×F

... ... ... ... ... ... ... ... ... .. ........ ..

µ

.....................................................................................................................

F

µ G × F ............................................. F e × id

. ......... .. ... ... ... F ...... ... ... ...

...... ..... ... .... .... . . .. .... ... ... .... ... . . F .... ... ....

id

F

We call (F, µ) a G stack. As usual, we’ll write g · x for µ(g, x). Then, the main result gives us the existence of a quotient stack. Theorem 5.2.1. [29] Let G be a flat, separated group scheme of finite presentation over S. Let F be a G- algebraic stack over S. Then there is a quotient stack [F/G], isomorphic to the stack of G-torsors (F/G)∗ and it is algebraic. We restate this for our purposes. Define [M 0,n (Pr × P1 , (d, 1))ss] as the substack of [M 0,n (Pr × P1 , (d, 1))] whose objects are semi-stable families (for some fixed line bundle such that the stable locus corresponds to the semi-stable locus.) Theorem 5.2.2. [[M 0,n (Pr × P1 , (d, 1))ss]/G] ∼ = [M 0,n (Pr , d)] as stacks. Proof. We first need to check that [M 0,n (Pr × P1 , (d, 1))ss] is indeed a Gstack. In our case, G is the constant sheaf. Take X ∈ S, g, h ∈ G and x ∈ Ob([M 0,n (Pr , d)ss ](X)). Recall that x = {π : C → X, {pi }1≤i≤n , µ : C → 99

Pr × P1 } is a family of stable maps over X. It’s clear that both of the above diagrams commute for objects. To see this for morphisms, take (α, β) : {π : C → X, {pi }, µ : C → Pr × P1 } → {π ′ : C ′ → X ′ , {p′i }, µ′ : C ′ → Pr × P1 }. Recall these are maps α : C → C ′ and β : X → X ′ such that the obvious maps commute. Then, g acts on (α, β) by taking it to (α, β) : {π : C → X, {pi }, g◦µ : C → Pr × P1 → Pr } → {π ′ : C ′ → X ′ , {pi }, g ◦ µ′ : C ′ → Pr × P1 } which again clearly commutes. Thus, by the Theorem, there is a quotient stack [[M 0,n (Pr , d)ss ]/G]. This is the stack associated to the prestack P, defined by Ob(P(T )) = Ob([M 0,n (Pr , d)ss](T )), and Mor(P(T )) = (f, g) where g ∈ G, and (f : g · x → y) ∈ Mor([M 0,n (Pr , d)ss ](T )). Finally, we check that [[M 0,n (Pr × P1 , (d, 1))ss]/G] ∼ = [M 0,n (Pr , d)]. We define a map u : [M 0,n (Pr , d)] → [[M 0,n (Pr × P1 , (d, 1))ss]/G] in the following way. • On objects, take x ∈ [M 0,n (Pr , d)](T ) considered as a map x : T → [M 0,n (Pr , d)] (See Lemma 5.1.1). Recall that [M 0,n (Pr , d)] → [M 0,n (Pr × P1 , (d, 1))ss] is a section given by attaching the identity morphism to P1 to the unique component C¯ ∈ C from Lemma 4.2.4. This extends to families, giving us a section. Compose the map x with this section to obtain a map x˜ : T → [M 0,n (Pr × P1 , (d, 1))ss]. Then our G equivariant map is µ ◦ (id × x˜) : G × T → [M 0,n (Pr × P1 , (d, 1))ss].

100

• On morhipsms, take x, y : T → [M 0,n (Pr , d)] ∈ Ob([M 0,n (Pr , d)](T )), and let (ϕ : x → y) ∈ Mor([M 0,n (Pr , d)](T )). The above section will send ϕ to the obvious map (ϕ : x˜ → y˜). This extends from the prestack to the stack. We need to show that u above is an equivalence of categories. Recall that locally essentially surjective means that when x ∈ Ob([M 0,n (Pr × P1 , (d, 1))ss/G](T )) that there is an element z ∈ Ob([M 0,n (Pr , d)](T )), such that there is a morphism u(z) → x. While u(z) may not equal x, they will only differ by an element of the group, which are exactly the morphisms in [M 0,n (Pr × P1 , (d, 1))ss]/G]. It’s clear that u sends Mor(x, x′ ) bijectively to Mor(u(x), u(x′ )). Remark 5.2.1. This argument works identically for our intermediate spaces from Chapter 3 as well. By applying the Theorem 1.1 of [17] we see that these stacks have a correpsonding coarse moduli space, namely, the intermediate spaces studied in Chapter 2.

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Vita

Adam Edgar Parker was born in Albany, New York on February 18, 1977 during a snowstorm to Mark Edgar Parker and Kristine Svitchan Parker. His hometown is Wadsworth, Ohio where he grew up with a younger sister (Dianna Jill) and brother (Jeffrey Milan). He attended Overlook and Isham Elementary Schools, Central Middle School, and Wadsworth High School, from which he graduated in 1995. He then left the state for the University of Michigan where he majored in Mathematics and Psychology. He graduated in May of 1999 and started graduate school at the University of Texas at Austin in the fall of that year.

Permanent address: 107 E. 48th St. Austin, Texas 78751

This dissertation was typeset with LATEX‡ by the author.

‡ A LT

EX is a document preparation system developed by Leslie Lamport as a special version of Donald Knuth’s TEX Program.

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