MODULI AND LIFTING OF ENRIQUES SURFACES

arXiv:1007.0787v1 [math.AG] 6 Jul 2010

CHRISTIAN LIEDTKE July 4, 2010

A BSTRACT. We construct the moduli space of Enriques surfaces in positive characteristic and determine its local and global structure. Also, we prove that Enriques surfaces lift to characteristic zero. The key ingredient is that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in P5 , even in characteristic 2.

I NTRODUCTION In order to give examples of algebraic surfaces with h1 (OX ) = h2 (OX ) = 0 that are not rational, Castelnuovo and Enriques constructed the first Enriques surfaces at the end of the 19th century. From the point of view of the Kodaira– Enriques classification, these surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. More precisely, these four classes consist of Abelian surfaces, K3 surfaces, Enriques surfaces and (Quasi-)Hyperelliptic surfaces. In characteristic 6= 2, Enriques surfaces behave extremely nice: deformations are unobstructed by results of Illusie [Il79] and Lang [La83]. Over the complex numbers, their moduli space is irreducible, smooth, unirational and 10-dimensional, and Kond¯o [Ko94] showed even rationality. Next, their fundamental groups are of order 2 and their universal covers are K3 surfaces. Moreover, Cossec [Co85] and Verra [Ve83] found explicit equations of these K3-covers: they are birational to complete intersections of three quadrics in P5 and the Z/2Z-action can be written down explicitly. (For generic Enriques surfaces this was already known to Enriques himself [En08].) Finally, Enriques surfaces in characteristic 6= 2 lift over the Witt ring, which is due to Lang [La83]. In characteristic 2, the situation is more complicated: first of all, as shown by Bombieri and Mumford [B-M76], Enriques surfaces fall into three classes, called classical, singular and supersingular. Although they still possess canonically defined flat double covers, which have trivial dualizing sheaves and which ”look“ cohomologically like K3 surfaces, these are in general only integral Gorenstein surfaces and may not be normal. It also happens that deformations are obstructed and finally, supersingular Enriques surfaces do not lift over the Witt ring. In this article, we clarify the situation in positive characteristic, and especially in characteristic 2. 2000 Mathematics Subject Classification. 14J28, 14J10. 1

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We start with the description of K3(-like) covers of Enriques surfaces. e → X be the K3(-like) cover of an Enriques surface X. Then Theorem. Let π : X there exists a morphism ϕ e → ϕ(X) e ⊆ P5 X e is a complete intersection of that is birational onto its image. The image ϕ(X) three quadrics. The exceptional locus of ϕ is, in a certain sense, a union of ADE-curves, see Theorem 3.1. Next, π is a torsor under a finite flat group scheme G of length 2. We describe the linear G-action on P5 induced by ϕ, and equations of the G-invariant e in Proposition 3.7. As a byproduct, we obtain quadrics cutting out ϕ(X)

Corollary. All Enriques surfaces in arbitrary characteristic arise via the Bombieri– Mumford–Reid construction in [B-M76, §3]. This result is the key to determining the local and global structure of the moduli space MEnriques of Enriques surfaces in positive characteristic p: Theorem. MEnriques is a quasi-separated Artin stack of finite type over k. (1) If p 6= 2, then MEnriques is irreducible, unirational, 10-dimensional and smooth over k. (2) If p = 2, then MEnriques consists of two irreducible, unirational and 10dimensional components Z/2Z

MEnriques

and

µ2 MEnriques .

Moreover, - they intersect along an irreducible, unirational and 9-dimensional α2 closed substack MEnriques , α2 - MEnriques parametrizes supersingular Enriques surfaces, µ2 - MEnriques − MEnriques is smooth and parametrizes singular surfaces, Z/2Z

Z/2Z

Z/2Z

- MEnriques − MEnriques parametrizes classical Enriques surfaces. It is smooth outside the locus of exceptional Enriques surfaces. Exceptional Enriques surfaces were introduced by Ekedahl and Shepherd-Barron in [E-SB04]. We note that Bombieri and Mumford conjectured, or, at least hoped for such a general picture already back in [B-M76] - except for the appearance of exceptional Enriques surfaces, on which we will comment below. To obtain these results, we first study polarized moduli, which turn out to be interesting in their own right, as they behave much nicer. In a certain sense, there is a class of natural and minimal polarizations for our setup, cf. the discussion at the beginning of Section 3, which is the following: Definition. A Cossec–Verra polarization on an Enriques surface X is an invertible sheaf L with self-intersection number 4 and such that every genus-one fibration |2E| on X satisfies L · E ≥ 2.

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In general, such invertible sheaves are not ample, but only big and nef. However, it is important to note that every Enriques surface possesses such a polarization. This is different from algebraic K3 surfaces, which are all polarizable, but where we need infinitely many types of polarizations to capture every one of them. Although every Enriques surface X possesses a Cossec–Verra polarization L, it is in general not unique. We refer to Proposition 3.4 for quantitative results. Contracting those curves that have zero-intersection with L, we obtain a pair (X ′ , L′ ), where X ′ is an Enriques surface with at worst Du Val singularities and L′ is an ample Cossec–Verra polarization. Theorem. Let X ′ be an Enriques surface over k with at worst Du Val singularities admitting an ample Cossec–Verra polarization. (1) If X ′ is not supersingular then it lifts over the√ Witt ring W (k). ′ (2) If X is supersingular then it lifts over W (k)[ 2], but not over W (k). Next, we denote by MCV,ample the moduli space of pairs (X, L), where X is an Enriques surface with at worst Du Val singularities and L is an ample Cossec–Verra polarization. This moduli space behaves extremely nice, even in characteristic 2: Theorem. MCV,ample is a quasi-separated Artin stack of finite type over k. (1) If p 6= 2 then MCV,ample is irreducible, unirational, 10-dimensional and smooth over k. (2) If p = 2 then MCV,ample consists of two irreducible, unirational, smooth and 10-dimensional components µ2 MCV,ample

and

Z/2Z

MCV,ample .

Moreover, - they intersect transversally along an irreducible, unirational, smooth α2 and 9-dimensional closed substack MCV,ample , α2 - MCV,ample parametrizes supersingular surfaces, α2 G − MCV,ample parametrizes singular surfaces (G = µ2 ) - MCV,ample and classical surfaces (G = Z/2Z), respectively, The lifting result and the description of MCV,ample give a beautiful picture of how Enriques surfaces ”should“ behave. However, we already mentioned above that MEnriques is not smooth along at points corresponding to classical and exceptional Enriques surfaces. Here is the reason: MCV,ample and MEnriques are related via simultaneous resolutions of singularities and then forgetting the polarization. It is Artin’s simultaneous resolution functor [Ar74b] which is responsible for singularities of MEnriques . This is similar to canonically polarized surfaces of general type, where Burns and Wahl [B-W74] showed how singularities of the canonical models may obstruct moduli spaces. We refer to Remark 5.9 for details. Because of these difficulties, we lose control over the ramification needed in order to lift Enriques surfaces. However, it suffices to prove the following: Theorem. Enriques surfaces lift to characteristic zero.

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We refer to Theorem 5.10 for what we know about ramification. Lifting over W (k) in characteristic p 6= 2, as well as for singular Enriques surfaces in p = 2 has been established by Lang [La83]. This article is organized as follows: After reviewing a couple of general facts in Section 1, we study projective and birational models of the canonical double cover of Enriques surfaces in Section 2. These results extend previous work of Cossec [Co85] to characteristic 2. The main difficulty is that Saint-Donat’s results [SD74] on linear systems on K3 surfaces cannot be applied to this double cover and we have to find rather painful ways around. In Section 3 we show that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in P5 . We introduce the notion of a Cossec–Verra polarization and establish a couple of general facts about them. Finally, we explicitly describe the action of the finite flat group scheme of length two, which acts on this complete intersection. In particular, we will see that every Enriques surface arises via the Bombieri–Mumford–Reid construction of [B-M76]. In Section 4 we study pairs of Enriques surfaces with at worst Du Val singularities together with ample Cossec–Verra polarizations. We prove their lifting to characteristic zero and construct the moduli space MCV,ample of such pairs. Using the results of Section 3, all boils down to describing deformations and moduli of complete intersections of three quadrics together with the action of a finite flat group scheme of length 2. Finally, in Section 5 we relate MCV,ample to the moduli space MEnriques of unpolarized Enriques surfaces. These are connected via Artin’s functor of simultaneous resolutions of singularities and the functor that forgets the Cossec–Verra polarization. Finally, we prove lifting to characteristic zero. Remark. While working on this article and discussing it, I was pointed out the discussion [E-S?] on the internet platform mathoverflow.net. It seems that some of the results in Section 5 have been obtained independently by Ekedahl and Shepherd-Barron several years ago, but were never published. Acknowledgements. I thank Brian Conrad, Igor Dolgachev, David Eisenbud and Jack Hall for long discussions and comments. Moreover, I gratefully acknowledge funding from DFG under research grant LI 1906/1-1 and thank the department of mathematics at Stanford university for kind hospitality. 1. G ENERALITIES We start by recalling a couple of general facts on Enriques surfaces, and refer to [B-M76] and [C-D89] for details and further references. Throughout this article, X denotes an Enriques surface over an algebraically closed field k of arbitrary characteristic p ≥ 0. By the definition of Bombieri and Mumford, this means ωX ≡ OX

and

b2 (X) = 10,

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where ≡ denotes numerical equivalence. Moreover, we have χ(OX ) = 1

In characteristic p 6= 2 we inequality h1 (OX ) ≤ 1 is true. sense to study the action of the either zero or a bijection.

and

b1 (X) = 0 .

have h1 (OX ) = 0, whereas for p = 2 only the Thus, if h1 (OX ) happens to be non-zero, it makes absolute Frobenius F on H 1 (OX ), which must be

Definition 1.1. An Enriques surface X is called ⊗2 ∼ (1) classical if h1 (OX ) = 0, hence ωX ∼ 6 OX and ωX = OX , = (2) singular if h1 (OX ) = 1, hence ωX ∼ = OX and F is bijective on H 1 (OX ), (3) supersingular if h1 (OX ) = 1, hence ωX ∼ = OX and F is zero on H 1 (OX ). The Picard scheme of X is smooth only if it is classical. More precisely, Picτ (X) is isomorphic to Z/2Z (classical), µ2 (singular) or α2 (supersingular), respectively. In each case, Picτ gives rise to finite and flat morphism of degree 2 e → X, π : X

which is a torsor under G := (Picτ (X))D , where −D = Hom(−, Gm ) denotes Cartier duality. In particular, if p 6= 2 or if X is a singular Enriques surface then G ∼ = Z/2Z, e the morphism π is an e´ tale Galois cover and X is a K3 surface. In the remaining e is never smooth, possibly even non-normal. cases, π is purely inseparable and X e In any case, X is an integral Gorenstein surface with invariants ωe ∼ = O e , χ(O e ) = 2 and h1 (O e ) = 0, X

X

X

X

i.e., ”K3-like“. Having only an integral Gorenstein surface rather than a smooth K3 surface as double cover, is one of the main reasons why Enriques surfaces in characteristic 2 are so difficult to come by. Finally, let us recall some of the Hodge invariants, see [La83, Theorem 0.11]: p 2

type classical singular supersingular

6= 2

h01 h10 h0 (ΘX ) h1 (ΘX ) h2 (ΘX ) 0 1 a 10 + 2a a 1 0 0 10 0 1 1 1 12 1 0 0 0 10 0

Classical Enriques surfaces satisfy a ≤ 1 and surfaces with a = 1 have been described and explicitly classified by Ekedahl and Shepherd-Barron [E-SB04] and Salomonsson [Sa03]. 2. P ROJECTIVE

MODELS OF THE DOUBLE COVER

In this section we study linear systems and projective models of the K3(-like) e of an Enriques surface X. Since there is no canonical polarization on X, e cover X the best thing to do is to consider pull-backs of invertible sheaves from X with positive self-intersection number. In characteristic 6= 2, this has been carried out by Cossec [Co85, Section 8]: such a pull-back defines a morphism that is either

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birational onto its image or generically finite of degree 2 onto a rational surface. Cossec’s proof relies on Saint-Donat’s analysis [SD74] of linear systems on K3 e In characteristic 2, the main difficulty is that X e is in surfaces that he applies to X. general only an integral Gorenstein surface, and so we have to take rather painful detours. The canonical double cover. We start by describing the canonical double cover of an Enriques surface X over a field k. Since Picτ (X) is a finite and flat group scheme of length 2, it gives rise to a torsor e → X π : X

under (Picτ (X))D , where −D denotes Cartier duality [Ra70, Proposition (6.2.1)]. More precisely, let P be the Poincar´e invertible sheaf on X × Picτ (X). By its universal property, there exists a morphism ψ : Picτ (X) → Picτ (X) such that P⊗P ∼ = (id × ψ)∗ P. Clearly, ψ = µ ◦ ∆, where ∆ is the diagonal and µ is the multiplication map of Picτ (X). Dualizing, we obtain an OX -algebra structure on P ∨ . Dualizing the multiplication map P ⊗ (OX ⊗ OPicτ (X) ) → P, we obtain a OX ⊗ OPicτ (X)D -comodule structure on P ∨ . Putting all this together, we obtain e ∼ π : X = Spec P ∨ → X

together with its (Picτ (X))D -action. In particular, this group scheme acts via its regular representation. Let us recall from [O-T70, Theorem 2] that a finite flat group scheme of length 2 over k is isomorphic to Ga,b for some a, b ∈ k with ab = 2. The assignment ρreg : Ga,b (S) = {s ∈ S | s2 = as} → GL2 (S) 1 s s 7→ 0 1 − bs

for any k-algebra S, defines the regular representation, see also [B-M76, §3] and Lemma 3.6 below. Projective models. As in [C-D89, Chapter III §2], we define Φ for an effective divisor C on X to be 1 Φ(C) := inf E · C, where |E| is a genus one pencil on X 2 For example, if C is an irreducible curve with C 2 > 0 then the linear system |C| is basepoint-free if and only if Φ(C) ≥ 2, see [C-D89, Theorem 4.4.1]. As we shall see now and in Theorem 2.4 below, Φ also controls the behavior of linear e systems on X. Theorem 2.1. Let C be an irreducible curve with C 2 > 0 and Φ(C) ≥ 2 on an Enriques surface X. Then (1) C 2 ≥ 4, e is globally generated, (2) the invertible sheaf π ∗ OX (C) on X ∗ (3) a generic Cartier divisor in |π OX (C)| is an integral Gorenstein curve, which is not hyperelliptic,

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(4) |π ∗ OX (C)| gives rise to a morphism

e → P(1+C 2 ) , ϕ : X

which is birational onto an integral surface of degree 2C 2 . P ROOF. By [C-D89, Theorem 4.4.1] the linear system |C| is base-point free. From the formula for h0 of [C-D89, Corollary 1.5.1] it follows that a generic divisor in |C| is reduced. Since C is irreducible by assumption, a generic divisor in |C| is reduced and irreducible. In particular, |C| has no fixed component. Now, if we had C 2 = 2 then |C| would define a morphism onto P1 , which contradicts C 2 6= 0 and we conclude C 2 ≥ 4. Moreover, since OX (C) is globally generated, e is also globally generated. it follows that π ∗ OX (C) on X Next, we consider the short exact sequence (1)

0 → OX (C) → π∗ π ∗ OX (C) → ωX (C) → 0 .

We have h1 (OX (C)) = 0 by [C-D89, Theorem 1.5.1], which, together with e π ∗ OX (C)) = 2 + C 2 . Thus, π ∗ OX (C) [C-D89, Corollary 1.5.1] implies h0 (X, e to (1+C 2 )-dimensional projective space. Also, gives rise to a morphism ϕ from X e is an since the image of |C| is a surface, the same is true for ϕ. Moreover, ϕ(X) e integral surface, i.e., reduced and irreducible, since X is. e is a smooth K3 surface If p 6= 2 or if X is a singular Enriques surface then X ∗ 2 2 ∗ and we compute (π C) = 2C . Since π OX (C) is globally generated, we find e Since non-degenerate and integral surfaces in PN have 2C 2 = deg ϕ · deg ϕ(X). degree at least N − 1, we conclude deg ϕ ≤ 2. If p = 2 and X is classical or supersingular, then π is a torsor under µ2 or α2 . In particular, π is inseparable and there exists a diagram (2)

X (1/2) 8J

88JJJ 88 J̟J 88 JJJ J% 88 8 e X F 88 88 88 π 8

ϕ

/

P1+C

2

X where F : X (1/2) → X denotes the k-linear Frobenius morphism. The composition ϕ ◦ ̟ corresponds to a linear subsystem of |2C| (here, we identify X with X (1/2) ). Both, ϕ and ̟ are morphisms, we have 2 deg ϕ = deg(ϕ ◦ ̟), as well as (2C)2 = 4C 2 . As before, we find deg ϕ ≤ 2, this time by arguing on X (1/2) . In order to show deg ϕ = 1 (again, for arbitrary π and p), we argue by contradiction similar to the proof of [Co85, Lemma 4.4.3]. So suppose deg ϕ 6= 1. e is an integral surface of degree C 2 in P1+C 2 , Then deg ϕ = 2 and the image ϕ(X) i.e., a surface of minimal degree. These surfaces have been explicitly classified by del Pezzo, see [E-H87] for a characteristic-free discussion. Now, the morphism π is a torsor under a finite flat group scheme G, which is e by G is isomorphic to X and not of length 2 over k. Since the quotient of X

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e it follows that the G-action on X e induces a non-trivial Gisomorphic to ϕ(X) e π ∗ OX (C))) and ϕ(X). e As already seen above, we may write action on P(H 0 (X, ∗ the global sections of π OX (C) as pr

e π ∗ OX (C)) → H 0 (X, ωX (C)) ω e → 0 . 0 → H 0 (X, OX (C)) → H 0 (X, X

Considered as a short exact sequence of G-modules, the G-action is trivial on H 0 (X, OX (C)), as well as H 0 (X, ωX (C)), but possibly not on ωXe . From the discussion at the beginning of this section it follows that H 0 (X, π∗ π ∗ L), as a Gmodule, decomposes into the direct sum of three regular representations of G, see also Proposition 3.7 below. We set P+ := P(H 0 (X, OX (C))), which is the projectivized +id-eigenspace for the G-action. If p 6= 2 or p = 2 and G ∼ = µ2 then the G-action has a second eigenspace, which we may identify with H 0 (X, ωX (C))ωXe . We denote by P− e π ∗ OX (C))) is fixed under the its projectivization. Clearly, if a point in P(H 0 (X, G-action (in the scheme-theoretic sense) then it lies in P+ or P− . For every v ∈ H 0 (X, ωX (C))ωXe , the hyperplane Pv := P(pr−1 (v)) is Ge in an irreducible curve ∆. stable and contains P+ . The generic Pv intersects ϕ(X) 2 2 C Since ∆ is of degree C − 1 in a P , it is a rational normal curve and in particular smooth and rational. Since ∆ is isomorphic to P1 and equipped with a non-trivial G-action, its fixed point scheme has length 2, which is supported in two distinct points if p 6= 2. e contains points that are fixed under G and so its intersection with Thus, ϕ(X) P+ or P− is non-empty. On the other hand, \ e = e P+ ∩ ϕ(X) {s = 0} ∩ ϕ(X) s∈π ∗ H 0 (X,OX (C))∨

e and s ∈ π ∗ H 0 (X, ωX (C))∨ . This implies that and similarly for P− ∩ ϕ(X) OX (C) or ωX (C) is not globally generated, a contradiction. This establishes deg ϕ = 1. By [Jo83, Th´eor`eme I.6.10], a generic Cartier divisor in |π ∗ OX (C)| is irree where ϕ ducible. The same theorem, applied to the open and dense subset of X e is an isomorphism and where X is smooth, shows that a generic Cartier divisor is generically reduced. Now, Cartier divisors on Gorenstein schemes are Gorenstein and in particular Cohen–Macaulay. Thus, a generic Cartier divisor in |π ∗ OX (C)| is irreducible, generically reduced and Cohen–Macaulay, i.e., irreducible and reduced, i.e., integral. e O e ) = 0, we conclude that ϕ induces Using the adjunction formula and H 1 (X, X on D the morphism associated to ωD . Thus, ϕ being birational, the generic D is not hyperelliptic. Hyperelliptic in the non-smooth case simply means that there exists a morphism of degree 2 onto P1 , see [Sch91]. Using Saint-Donat’s analysis [SD74] of linear systems on K3 surfaces, Cossec e in characteristic 6= 2 is cut out by quadrics: [Co85] has shown that ϕ(X)

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e Theorem 2.2 (Cossec+ε). Under the assumptions of Theorem 2.1, the image ϕ(X) 2 in P1+C is projectively normal and cut out by quadrics, whenever (1) char(k) 6= 2, or (2) char(k) = 2 and X is a singular Enriques surface. P ROOF. In characteristic 6= 2, this is shown in [Co85, Section 8]. e is smooth, ϕ is If X is a singular Enriques surface in characteristic 2, then X e has only isolated singularities. But then, a generic divisor birational and ϕ(X) ∗ D ∈ |π OX (C)| is smooth and the whole analysis in [SD74, Section 7] remains valid also in characteristic 2. Thus, [SD74, Theorem 7.2] and [Co85, Lemma 8.1.2] e is projectively normal and cut out by quadrics. show that ϕ(X) e is always cut out by quadrics – this would follow from It is plausible that ϕ(X) e (we refer to [CFHR, Section 3] for numerical 4-connectedness of π ∗ OX (C) on X a discussion of this notion for singular varieties). However, we have only been able to establish this in the special case, in which we are interested in later on: Proposition 2.3. In addition to the assumptions of Theorem 2.1, assume that C 2 = e ⊂ P5 is projectively normal and cut out by quadrics. 4 and Φ(C) = 2. Then ϕ(X)

e is P ROOF. We have to show that the graded ring associated to π ∗ OX (C) on X generated in degree 1 with relations in degree 2 only, i.e., a Koszul algebra. By Theorem 2.1, a generic Cartier divisor D ∈ |π ∗ OX (C)| is an integral and non-hyperelliptic Gorenstein curve of arithmetic genus pa (D) = 5. For n ≥ 1, we e consider the following short exact sequences on X: ⊗n → 0. 0 → π ∗ OX ((n − 1)C) → π ∗ OX (nC) → ωD

Pushing π ∗ OX ((n − 1)C) forward to X and using [C-D89, Theorem 1.5.1], we e π ∗ OX ((n − 1)C)) = 0 for n ≥ 1. Thus, as explained in the proof conclude h1 (X, of part (ii) of [SD74, Theorem 6.1], to prove our assertion, it suffices to show that the canonical ring of D is a Koszul algebra. Before proceeding, we study genus one half-pencils on X more closely: since Φ(C) = 2, there exists a genus one pencil E on X with C · E = 4. Moreover, let E ′ be a genus one curve with |2E ′ | = |E|, i.e., a half-pencil. We now claim: (1) π ∗ OX (E ′ ) is globally generated with h0 = 2 and h1 = 0, and (2) π ∗ OX (C − E ′ ) also satisfies h0 = 2 and h1 = 0. It is globally generated outside π −1 (R), where R is a (possibly empty) union of ADE-curves. We only deal with the case that π is inseparable in characteristic 2 and leave the remaining and easier cases to the reader: the assertions about h0 and h1 of π ∗ OX (E ′ ) follow from pushing it down to X and then using h0 (X, OX (E ′ )) = h0 (X, ωX (E ′ )) = 1, as well as h1 (X, OX (E ′ )) = h1 (X, ωX (E ′ )) = 0. Now, we use diagram (2): the linear system ̟ ∗ |π ∗ OX (E ′ )| is a linear subsystem of |̟ ∗ π ∗ OX (E ′ )| = |F ∗ E ′ | = |2E ′ | = |E|. Since both satisfy h0 = 2, they are equal and so π ∗ OX (E ′ ) is globally generated since OX (E) is. Let us adjust the proof of [Co85, Theorem 5.3.6] to our situation: by Riemann-Roch there exists an effective divisor D such that E ′ + D ∈ |C|, E ′ D = 2, and D 2 = 0. Moreover,

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there exists a divisor E ′′ of canonical type such that D = E ′′ + R with R ≥ 0. Since Φ(C) = 2 we have CE ′′ ≥ 2 and if equality holds then E ′′ is a genus one half-pencil. Thus, 4 = C 2 ≥ CE ′ + CE ′′ ≥ 4, and we conclude CE ′′ = 2 and CR = 0. In particular, if non-empty, R is a union of ADE-curves. The remaining assertions now follow as before, establishing our two claims. Next, let us show that ϕ(D) possesses a simple (pa (D) − 2)-secant: first, we e π ∗ OX (C −E ′ )) = choose a generic Cartier divisor G ∈ |π ∗ OX (E ′ )|. Since h1 (X, 0 ∗ 0 e π OX (C)) surjects onto H (G, π ∗ OX (C)|G ). Using 0, we conclude that H (X, deg π ∗ OX (C)|G = 4 we see that ϕ embeds G as a quartic into some P3 , which is easily seen to be the complete intersection of two quadrics. Thus, a generic hyperplane H of P5 intersects this complete intersection in 4 points in uniform position. e an integral curve ϕ(D), where D ∈ |π ∗ OX (C)|, having This H cuts out on ϕ(X) the stated simple (pa (D) − 2)-secant. Having established this (pa (D) − 2)-secant, [Sch91, Theorem 1.2] and [Sch91, Corollary 1.3] show that the canonical ring of D is generated in degree 1 and has relations in degree ≤ 3. Our proposition is proved once we show that no relations in degree 3 are needed. Suppose that relations in degree 3 are needed. Then, ϕ(D) is contained in an irreducible surface S of degree pa (D)−2 by [Sch91, Theorem 3.1]. By the classification of surfaces of minimal degree [E-H87] together with pa (D) = 5 we find that S is ruled. Moreover, a generic ruling of S intersects ϕ(D) in three distinct smooth points. Thus, D possesses a globally generated invertible sheaf M of degree 3 with h0 = 2 (a “g31 ”) and D is trigonal. Now, we consider L := π ∗ OX (E ′ )|D . Then π ∗ OX (C − E ′ )|D ∼ = ωD ⊗ L∨ and L and ωD ⊗ L∨ are invertible sheaves of degree 4 = 21 deg ωD . Taking cohomology in 0 → π ∗ OX (E ′ − C) → π ∗ OX (E ′ ) → L → 0, e π ∗ OX (E ′ )) and H 0 (X, e π ∗ OX (C − we obtain h0 (D, L) ≥ 2. Moreover, H 0 (X, ′ 0 0 ∨ E )) inject into H (D, L) and H (D, ωD ⊗ L ), respectively. In particular, L is globally generated since π ∗ OX (E ′ ) is. Choosing D generically, we may assume that D does not intersect π −1 (R) and then ωD ⊗ L∨ is globally generated. Since h1 (D, L) 6= 0, Clifford’s inequality implies h0 (D, L) ≤ 3 and equality could only happen if D were hyperelliptic. Thus, h0 (D, L) = h0 (D, ωD ⊗ L∨ ) = 2. This is enough to show that D is not trigonal: by [ACGH, Excercise III.B-5], which works for integral Gorenstein curves in arbitrary characteristic, the invertible sheaf M making D trigonal would have to be a subsheaf of L or ωD ⊗ L∨ , which is absurd. e arising from curves We complete the picture by discussing linear systems on X C on X with Φ = 1: Theorem 2.4. Let C be an irreducible curve with C 2 > 0 and Φ(C) = 1 on an Enriques surface X. Then e is globally generated, (1) the invertible sheaf π ∗ OX (C) on X

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(2) |π ∗ OX (C)| gives rise to a morphism e → P(1+C 2 ) , ϕ : X

which is generically of degree 2 onto a surface of minimal degree C 2 , e is cut out by quadrics. (3) the image ϕ(X)

e π ∗ OX (C)) = 2 + C 2 . P ROOF. As in the proof of Theorem 2.1 we find h0 (X, ∗ Again, |C| has no fixed component and so π OX (C) is globally generated outside a finite (possibly empty) set of points. e is a surface since the image of Let us first assume C 2 ≥ 4. In this case, ϕ(X) the rational map associated to |C| is a surface [C-D89, Theorem 4.5.1]. Seeking a contradiction, we assume that ϕ is birational. As in the proof of Theorem 2.1, we conclude that a generic Cartier divisor D ∈ |π ∗ OX (C)| is an integral Gorenstein curve. Since Φ(C) = 1, there exists a genus one half-pencil E ′ on X such that C · E ′ = 1. Then L := π ∗ OX (E ′ )|D satisfies deg L = 2 and taking cohomology in 0 → π ∗ OX (E ′ − C) → π ∗ OX (E ′ ) → L → 0 we find h0 (D, L) ≥ 2. Since pa (D) ≥ 5, Riemann-Roch implies h1 (D, L) 6= 0. But then, Clifford’s inequality h0 (D, L) ≤ 2 is in fact an equality, which implies that D is hyperelliptic. In the proof of Theorem 2.1 we have seen that ϕ restricted to D induces |ωD |, which contradicts the fact that ϕ is birational. Thus, deg ϕ ≥ 2 e is a non-degenerate integral surface in P1+C 2 , we conclude and since ϕ(X) e 2C 2 ≤ deg ϕ · C 2 ≤ deg ϕ · deg ϕ(X).

On the other hand, π ∗ OX (C) is globally generated outside a finite set of points and so we find e ≤ 2C 2 deg ϕ · deg X

with equality if and only if π ∗ OX (C) is globally generated: this is clear if π is e is smooth. If π is inseparable, we consider ϕ ◦ ̟ in the e´ tale, because then X diagram (2) and obtain the same result by arguing on X (1/2) . Putting these inequalities together, we find that π ∗ OX (C) is globally generated, e = C 2 . In particular, ϕ(X) e is a surface of minimal degree deg ϕ = 2 and deg ϕ(X) and thus cut out by quadrics [E-H87]. It remains to deal with the case C 2 = 2. Then, ϕ is a possibly rational map to 3 e is a curve. A generic G ∈ |π ∗ OX (E ′ )|, P . By contradiction, assume that ϕ(X) ′ ′ where E is a half-pencil with C · E = 1, is an integral curve with pa = 1. We find deg π ∗ OX (C)|G = 2, which implies h0 (G, π ∗ OX (C)|G ) = 2 by Riemann-Roch and Clifford’s inequality. This implies that ϕ(G) is a linearly embedded P1 ⊂ P3 . e is equal to this P1 , contradicting that ϕ(X) e linearly spans P3 . Thus, But then ϕ(X) e is a surface and we conclude as before. ϕ(X)

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CHRISTIAN LIEDTKE

3. C OMPLETE

INTERSECTIONS OF THREE QUADRICS

e which turns out to In this section we study one particular birational model of X, 5 be the complete intersection of three quadrics in P . This extends results of Cossec [Co85, Section 8] and Verra [Ve83, Theorem 5.1] to characteristic 2. We explicitly e As describe the equations and the action of the finite flat group scheme G on X. a byproduct, we obtain that all Enriques surfaces in any characteristic arise via the Bombieri–Mumford–Reid construction in [B-M76, §3]. e we study linear systems |π ∗ OX (C)|, In order to find projective models of X, 2 where C is an irreducible curve with C > 0. By Theorem 2.1 and Theorem 2.4, e → PN is birational onto its image if and only the associated morphism ϕ : X 2 if Φ(C) ≥ 2 and C ≥ 4. In this case, the codimension of the image is equal to C 2 − 1. Thus, in our setup, models of smallest possible codimension are of codimension 3 in P5 . Moreover, by [C-D89, Lemma 3.6.1], irreducible curves with C 2 < 10 satisfy Φ(C) ≤ 2. We are thus led to studying irreducible curves with C 2 = 4 and Φ(C) = 2. e → P5 , Theorem 3.1. For every Enriques surface X there exists a morphism ϕ : X which is birational onto its image. More precisely, there is a Cartesian diagram ϕ

:

/ ϕ(X) e

e X

π

X

/ P5

π′

ν

/ X′

such that e is a complete intersection of three quadrics, (1) ϕ(X) (2) ν is a birational morphism and X ′ has at worst Du Val singularities, (3) π is a torsor under under a finite flat group scheme G, which arises as e → X ′ , and pull-back from a G-torsor ϕ(X) e is induced by a linear G-action of the ambient P5 . (4) the G-action on ϕ(X)

P ROOF. Let C be an irreducible curve on X with C 2 = 4 and Φ(C) = 2, which always exists by [C-D89, Chapter IV §9], but see also Proposition 3.4. We set L := OX (C). By Theorem 2.1 and Proposition 2.3, the invertible sheaf π ∗ L on e gives rise to a birational morphism, whose image ϕ(X) e is cut out by quadrics. X 0 ∗ 0 ∗ ⊗2 e e We compute h (X, π L) = 6 and h (X, π L ) = 18 via pushing forward these sheaves to X (see the proof of Theorem 2.1 details). Thus, there are three quadric e is a complete intersection of three quadrics. relations and hence ϕ(X) e induces a G-action on H 0 (X, e π ∗ L). This gives rise to Next, the G-action on X e a linear G-action on P5 extending the G-action on ϕ(X). Every irreducible curve that has zero-intersection with C is a (−2)-curve [C-D89, Proposition 4.1.1]. Since OX (C) is globally generated, big and nef, M ν : X → X ′ := Proj H 0 (X, L⊗n ) n≥0

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13

is a birational morphism that contracts those (−2)-curves having zero-intersection with C and nothing else. In particular, X ′ has at worst Du Val singularities. Thus, H 1 (X, OX ) ∼ = H 1 (X ′ , OX ′ ) and ωX is 2-torsion if and only if ωX ′ is. This e ′ → X ′. implies that the canonical G-torsor π arises as pull-back from a G-torsor X ′ ⊗n ⊗n Since X has only Du Val singularities, L (resp. ωX ⊗ L ) for n ≥ 0 and ν∗ (L⊗n ) (resp. ν∗ (ωX ⊗ L⊗n )) have isomorphic global sections. Thus, the graded e ′ ′∗ eπ∗ L of (X, e π ∗ L) is isomorphic to the graded ring R e′ ′∗ ring R π ν∗ L of (X , π ν∗ L). eπ∗ L , O(1)) is just ϕ(X) e ⊂ P5 by Proposition 2.3. On the other Now, (Proj R ′ hand, ν∗ L is ample on X (by the Nakai–Moisehzon criterion, see also the proof e ′ . Thus, X e ′ is of Proposition 3.4 below), and so ν∗ π ∗ L ∼ = π ′∗ ν∗ L is ample on X ′ e ∗ . isomorphic to Proj R π L e it is natural to Polarizations. Having just established a projective model of X, 5 e → P , as well as ’how far’ ϕ is from being an ask for uniqueness of ϕ : X isomorphism. Since our previous result extends work of Cossec [Co85, Section 8] and Verra [Ve83, Theorem 5.1] to characteristic 2, we define Definition 3.2. A Cossec–Verra vertible sheaf L ∈ P(X), where P(X) := L ∈ Pic(X)

polarization on an Enriques surface X is an in there exists an irreducible curve C . with C ∈ |L|, C 2 = 4, Φ(C) = 2

e → P5 corresponding to |π ∗ L| by ϕL . We denote the morphism X

Clearly, every L ∈ P(X) is big and nef. To decide whether it is ample, let us recall that an irreducible curve on X is a (−2)-curve, i.e., has self-intersection −2, if and only if it is smooth and rational. Such curves are called nodal and we denote by R(X) the set of all nodal curves. Proposition 3.3. For L ∈ P(X) the following properties are equivalent: (1) L · α > 0 for every α ∈ R(X), (2) L is ample, (3) π ∗ L is very ample, and e → ϕL (X) e is an isomorphism. (4) ϕL : X In general, the reduced exceptional locus of ϕL is the union of the reduced inverse images of all those nodal curves having zero-intersection with L. P ROOF. By [C-D89, Corollary 3.2.2], every effective divisor is linearly equivalent to one that is the positive sum of curves of arithmetic genus 1 and 0. Since Φ(C) = 2, the intersection of L with curves of arithmetic genus 1 is positive. Thus, by the Nakai–Moishezon criterion for ampleness, L is ample if and only if it has positive intersection with every nodal curve. This establishes 1 ⇔ 2. From the proof of Theorem 3.1 we get 1 ⇔ 4. The equivalence 3 ⇔ 4 is obvious. The last assertion follows from the proof of Theorem 3.1. An Enriques surface X is called unnodal if R(X) is empty. Over the complex numbers, a generic Enriques surface is unnodal [B-P83, Proposition 2.8]. Let us

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CHRISTIAN LIEDTKE

recall that a Reye congruence is the subset of G(1, 3) parametrizing lines in P3 that lie on at least two quadrics of a given generic 3-dimensional family of quadrics. By [Co83, Theorem 1], a generic nodal Enriques surfaces is a Reye congruence. This said, we establish a couple of facts about existence, uniqueness and ampleness of Cossec–Verra polarizations: Proposition 3.4. Cossec–Verra polarizations always exist. More precisely: (1) if X is unnodal, then P(X) is infinite and every L ∈ P(X) is ample, (2) if X is a generic Reye congruence and p 6= 2 then there exists an ample Cossec–Verra polarization. Moreover, if p ≥ 19 then P(X) is infinite. (3) if X is extra special then P(X) is finite and no Cossec–Verra polarization is ample. Moreover, there exist nodal curves on X, whose inverse images e are contracted by ϕL for every L ∈ P(X). on X Over the complex numbers, P(X) modulo Aut(X) is finite, and is of order 252, 960 for a generic Enriques surface. Remark 3.5. Extra special surfaces exist in characteristic 2 only and are discussed e8 -extra special surface possesses only in [C-D89, Chapter III.5]. For example, an E e is non-normal and the only ϕL partially conone Cossec–Verra polarization, X tracts the non-normal locus. We shall see in Section 5 below that this is closely related to obstructions of the deformation functor, as well as to the exceptional Enriques surfaces studied in [E-SB04]. P ROOF. Let us recall, e.g. from [C-D89, Chapter II.5] that the Enriques lattice E, i.e., the N´eron–Severi group of an Enriques surface modulo torsion, is isometric to the hyperbolic lattice corresponding to the Dynkin diagram T2,3,7 : α1

α2

α3

α4

α5

α6

α7

α8

α9

◦

◦

◦

◦

◦

◦

◦

◦

◦

◦

α0

Let WE be the Weyl group with respect to all roots and WX be the Weyl group with respect to R(X). We denote by CX := {x ∈ E | xα ≥ 0, ∀α ∈ R(X)} the nodal chamber of X. As explained in [C-D89, Chapter III.2], CX is a fundamental domain of VX := {x ∈ E | x2 ≥ 0} for the WX -action. Moreover, let VX+ be the connected component of VX containing the class of an ample divisor and set + := VX+ ∩ CX . CX ∨ Let ω1 := α∨ 1 ∈ E be the fundamental weight of the root α1 ∈ E. It follows from [C-D89, Corollary 2.5.7] that elements of P(X) correspond to those elements + . In particular, P(X) is not empty. The of the orbit Isom(E) · ω1 that lie in CX stabilizer Stab(ω1 ) is the Weyl group corresponding to the Dynkin diagram T2,3,7 with vertex α1 removed, which is of type D9 . In particular, Stab(ω1 ) is finite.

ENRIQUES SURFACES

15

+ = VX+ and so P(X) corresponds to If X is unnodal then WX is trivial, CX the cosets of WE = Isom(E)/{±id} modulo Stab(ω1 ), which is infinite. Since X is unnodal, every L ∈ P(X) is ample by Proposition 3.3. Moreover, from Stab(ω1 ) ∼ = W (D9 ), we infer W (D9 )/W (D9 )(2) ∼ = (Z/2Z)8 ⋊ S9 , see [C-D89, Proposition 2.8.4]. By [B-P83, Theorem (3.4)], a generic complex Enriques surface satisfies Aut(X) ∼ = WE (2) and thus WE /WE (2) ∼ = O + (10, F2 ) by [C-D89, Theorem 2.9.1]. This identifies P(X) modulo Aut(X) with WE /WE (2) modulo W (D9 )/W (D9 )(2), which has

220 · 35 · 52 · 7 · 17 · 31 |O+ (10, F2 )| = = 25 · 3 · 5 · 17 · 31 = 252, 960 |(Z/2Z)8 ⋊ S9 | 215 · 34 · 5 · 7 elements. Next, let X be a generic Reye congruence. It contains 10 genus-one half pencils Fi and 10 nodal curves Di such that Fi Fj = 1 and Di Dj = 2 for i 6= j, see [Co83, Lemma 3.2.1]. It follows from the proof of [Co83, Proposition 3.2.5], that the invertible sheaf corresponding to Cij := Fi + 21 (Di + Dj ) for i 6= j belongs to P(X). If X is a generic Reye congruence then the genus-one fibrations |2Fi | have no reducible fibers by the remark after [Co83, Proposition 3.2.4]. From this it is not difficult to compute that every nodal curve intersects Cij positively, i.e., the corresponding ϕL is an isomorphism. It follows from [C-D85, Theorem 1], that automorphism groups of generic nodal Enriques surfaces in characteristic p ≥ 19 + and since Stab(ω1 ) is finite, we conclude are infinite. Since this group acts on CX that P(X) is infinite. e8 -extra special, then WX = Isom(E)/h±idi, which implies that it If X is E contains only one genus-one fibration |2E| and that P(X) consists of only one element L. Then we use [C-D89, Proposition 3.6.2] to see that |L| = |2E + 2R1 + ... + 2R7 + R8 + R10 | (notation as in case 4 of [C-D89, page 185]) from which we read off that R1 , ..., R8 and R10 have zero-intersection with L. In the other extra special cases, the genus one fibrations are described in [C-D89, Chapter III.5] and applying [C-D89, Proposition 3.6.2] to a divisor class |C| with C 2 = 4, Φ(C) = 2 we end up with a finite list of possibilities of how to write |C| in terms of genus-one fibrations. First, this shows that P(X) is finite. Second, in these explicit lists we can always find nodal curves that have zero-intersection with C for any choice of C and any decomposition into genus-one pencils. We leave the lengthy, yet straight forward details to the reader. Finally, over the complex numbers, the subgroup of Isom(E) generated by WX and Aut(X) is of finite index [Do84]. In particular, there are only finitely many + ) and orbits of Isom(E) · ω1 modulo WX (needed to move the vector into CX modulo Aut(X). Thus, P(X) modulo Aut(X) is finite. Explicit equations. We end this section by determining explicit equations of the e For later use, let us extend our setup for a moment: complete intersection ϕL (X). let R be a complete, local and Noetherian ring with residue field k. Then, a finite flat group scheme of length 2 over R is isomorphic to Ga,b for some a, b ∈ R

16

CHRISTIAN LIEDTKE

with ab = 2 by [O-T70, Theorem 2]. Straight forward calculations – but see also [B-M76, p.222] – show: Lemma 3.6. The regular representation of Ga,b , where ab = 2, associates to every R-algebra S the homomorphism ρreg : Ga,b (S) = {s ∈ S | s2 = as} → GL2 (S) 1 s s 7→ 0 1 − bs

If we set T := R[x1 , y1 , ..., xn , yn ] and assume that Ga,b acts on each pair xi , yi via ρreg then the following quadrics are Ga,b -invariant xi xj ,

yi2 − a xi yi ,

xi y j + y i xj + b y i y j .

Moreover, the Ga,b -invariants of T in even degree are generated by these invariant quadrics. τ ∼ Let X be an Enriques surface over k and assume that Pic (X) = Gb,a . At the beginning of Section 2 we gave an explicit description of the induced Ga,b -torsor e → X. π : X

Next, we choose a Cossec–Verra polarization L on X, and remind the reader that such polarizations always exist by [C-D89, Chapter IV §9] or Proposition 3.4. Proposition 3.7. There exists a linear Ga,b -action on P5 such that e → P5 , ϕL : X

becomes Ga,b -equivariant. Its image is the complete intersection of three quadrics. More precisely, there exist coordinates x1 , x2 , x3 , y1 , y2 , y3 on P5 such that (1) the Ga,b -action on each pair xi , yi is as in Lemma 3.6, and e are linear combinations of the (2) such that the quadrics cutting out ϕ(X) invariant quadrics of Lemma 3.6. e is a complete intersection of three P ROOF. By Theorem 3.1, the image ϕ(X) quadrics. We take cohomology in the short exact sequence 0 → L → π∗ π ∗ L → ω X ⊗ L → 0

and note that Ga,b acts via its regular representation on π∗ π ∗ L, see the discussion at the beginning of Section 2. Thus, we can choose a basis x1 , x2 , x3 of H 0 (X, L), as well as lifts y1 , y2 , y3 of a basis of H 0 (X, ωX ⊗L) to H 0 (X, π∗ π ∗ L) such that Ga,b acts on each pair xi , yi as in Lemma 3.6. In particular, as a Ga,b -representation, H 0 (X, π∗ π ∗ L) is isomorphic ρ⊕3 reg , that is, 3 copies of the regular representation. Now, consider the exact sequence of Ga,b -modules µ

0 → ker µ → Sym2 H 0 (X, π∗ π ∗ L) → H 0 (X, π∗ π ∗ (L⊗2 )) → 0 .

The kernel ker µ is easily seen to be 3-dimensional. Arguing as above, we see that H 0 (X, π∗ π ∗ L⊗2 ) is isomorphic to ρ⊕9 reg as Ga,b -representation. Decomposing the Ga,b -representation on Sym2 H 0 (X, π∗ π ∗ L), we find that Ga,b acts trivially on e is cut out by three quadrics, all of which are Ga,b -invariant. ker µ. Thus, ϕ(X)

ENRIQUES SURFACES

17

Remark 3.8. Following an idea of Reid, Bombieri and Mumford [B-M76, §3] gave the first construction of all three types of Enriques surfaces in characteristic 2. Our result shows that in fact all Enriques surfaces arise in this way - after possibly resolving Du Val singularities of the quotient. 4. M ODULI

AND

L IFTING –

THE

P OLARIZED C ASE

In this section we consider Enriques surfaces together with ample Cossec–Verra polarizations. Given an Enriques surface, such a polarization always exist after possibly contracting nodal curves to Du Val singularities. Thus, we study pairs (X, L), where X is an Enriques surface with at worst Du Val singularities and L is an ample Cossec-Verra polarization. We show that such pairs have an extremely nice deformation theory, construct their moduli space MCV,ample and prove lifting to characteristic zero. Let us first slightly extend Definition 3.2: an invertible sheaf L on an Enriques surface X ′ with at worst Du Val singularities is a Cossec–Verra polarization if ν ∗ L on X is, where ν : X → X ′ denotes the minimal desingularization. In particular, if L is a Cossec–Verra polarization on a (smooth) Enriques surface then M ν : X → X ′ := Proj H 0 (X, L⊗n ) n≥0

is a contraction and OX ′ (1) is an ample Cossec–Verra polarization on X ′ . In this section, k denotes an algebraically closed field of characteristic p ≥ 0 and R is a complete, local and Noetherian ring with residue field k. Picard scheme and effectivity. Before studying deformations, moduli and lifting, we have to understand extensions of invertible sheaves on formal deformations. Proposition 4.1. Let X be an Enriques surface with at worst Du Val singularities over k and X → Spf R be a formal deformation of X. If L is an invertible sheaf on X and (1) if X is classical, then L extends uniquely to X , and (2) if X is non-classical, then L⊗2 extends to X . Moreover, if L1 and L2 are ⊗2 ⊗2 extensions of L to R then L1 ∼ = L2 . In particular, every formal deformation is effective. P ROOF. If X is classical then h1 (OX ) = h2 (OX ) = 0 and so the first assertion is a standard result in deformation theory. If X is non-classical then char(k) = 2. We write X as limit Xn → Spec Rn with X = X0 and where each Rn+1 → Rn is a small extension. Thus, for every n ≥ 0 we have an exponential sequence × × → 0, → OX 0 → OX → OX n n+1

and, taking successive extensions, we end up with short exact sequences (3)

× × → 0. → OX 0 → Kn,s → OX n n+s

18

CHRISTIAN LIEDTKE

We claim that Kn,s , considered as sheaf of Abelian groups, is a direct sum of Witt vectors of finite length over OX : first of all, we have Kn,1 ∼ = OX for all n. Moreover, interpreting Xn+s as X(n+1)+(s−1) , the respective sequences (3) yield a short exact sequence 0 → Kn+1,s−1 → Kn,s → OX → 0 .

By induction on s we may assume that Kn+1,s−1 is a direct sum of sheaves of Witt vectors of finite length. Considering each of its summand individually, Lemma 4.3 reveals that also Kn,s is of this form. This establishes our claim. × ). Taking cohomology Now, L corresponds to an element in Pic(X) ∼ = H 1 (OX 2 in (3), we denote by δs the coboundary map Pic(X) → H (K0,s ). Then δs (L) 6= 0 if and only if L extends to Xs . Since K0,s is a direct sum of Wm OX ’s and since H 2 (Wm OX ) ∼ = k for every m by Lemma 4.3, we conclude that H 2 (K0,s ) is 2-torsion. Thus, for every s, the obstruction to extending L⊗2 to Xs vanishes, proving that L⊗2 extends to X . Moreover, if L1 and L2 are extensions of L to Xs , then their “difference” lies in H 1 (K0,s ), which is 2-torsion by the same reasoning as for H 2 (K0,s ) above. This ⊗2 ⊗2 implies L1 ∼ = L2 . Finally, if L is ample on X, then so is L⊗2 and since the latter extends to X , the formal deformation is effective by Grothendieck’s existence theorem. Remark 4.2. The classical case is trivial. In case X is singular and R = W (k), extension of L⊗4 has been shown in [La83, Theorem 1.4]. Lemma 4.3. Let X be a non-classical Enriques surface over k. Then Ext1 (OX , Wm OX ) ∼ and H 2 (Wm OX ) ∼ = H 1 (Wm OX ) ∼ = k = k.

Moreover, the only non-trivial extension class corresponds to Wm+1 OX . P ROOF. We have H 1 (W OX ) = 0 and H 2 (W OX ) ∼ = k by [Il79, Section II.7.3]. Thus, the m-fold Verschiebung V m yields an exact sequence Vm

0 → H 1 (Wm OX ) → H 2 (W OX ) → H 2 (W OX ) → H 2 (Wm OX ) → 0. Since H 1 (Wm OX ) ∼ = Ext1 (OX , Wm OX ) is contained in k and contains the nontrivial class [Wm+1 OX ], we conclude H 1 (Wm OX ) ∼ = k for all m. In particular, this class generates the cohomology group. Counting dimensions in the above exact sequence, we infer H 2 (Wm OX ) ∼ = k for all m.

Proposition 4.4. Under the previous assumptions, Picτ (X /R) is a finite flat group scheme of length 2 over R.

P ROOF. By [B-M76, Theorem 2], Picτ of an Enriques surface over any field is finite of rank 2, and so Picτ (X /R) is quasi-finite. Choosing an ample invertible sheaf L on X and extending L⊗2 to X , which is possible by our previous result, we see that X → Spec R is projective. Thus, R is Noetherian, and X → Spec R is smooth, projective and with geometrically irreducible fibers, which implies that Picτ (X /R) is projective [FGA, no 236, Corollaire 4.2]. Thus, Picτ (X /R) is in fact a finite group scheme over R. As Picτ of the special fiber has rank 2, Nakayama’s

ENRIQUES SURFACES

19

lemma implies that the R-module H 0 (Spec R, Picτ (X /R)) is generated by 2 elements. If ωX 6∼ = OX then these two invertible sheaves define two distinct morphisms from Spec R to Picτ (X /R). Thus, the closure of the union of their images defines a non-trivial finite flat subgroup scheme. By the previous discussion this group scheme has to coincide with Picτ (X /R). We may thus assume ωX ∼ = OX . Let us write X as limit over Xn → Rn , where the Rn ’s are local Artin algebras. Clearly, we have f∗ OXn = Rn and using Grothendieck duality, we find R2 f∗ OXn ∼ = Rn . By flatness, we = R 2 f ∗ ω Xn ∼ P i i have i (−1) length(R f∗ OXn ) = χ(OX ) = 1. Thus, the length of R1 f∗ OXn as Rn -module is equal to the length of Rn as Rn -module. From H 1 (OX ) ∼ = k and 1 Nakayama’s lemma we find that R f∗ OXn is a cyclic Rn -module. This implies that R1 f∗ OXn ∼ = Rn . Now, R1 f∗ OXn is the tangent space to Pic(Xn /Rn ) at the zero-section. Since this is locally free of rank 1, we infer the existence of a finite flat and infinitesimal group scheme of rank 2 inside Pic0 (Xn /Rn ). Since this holds for arbitrary n, we conclude that the limit of these group schemes coincides with Picτ (X /R). Let us recall once more that every finite flat group scheme of rank 2 over R is of the form Ga,b for some a, b ∈ R with ab = 2, see [O-T70, Theorem 2], as well as Lemma 3.6. The following result extends Proposition 3.7 to families: Proposition 4.5. Let L be a Cossec–Verra polarization on X and assume that it extends to X → Spec R. If Picτ (X /R) ∼ = Gb,a , then there exists a linear Ga,b action on P5R , as well as a Ga,b -equivariant morphism ϕ : Xe → P5R ,

whose image is the complete intersection of three quadrics. Moreover, there exist coordinates x1 , x2 , x3 , y1 , y2 , y3 on P5R such that (1) the Ga,b -action is as in Lemma 3.6, and (2) such that the quadrics cutting out ϕ(Xe) are R-linear combinations of the invariant quadrics of Lemma 3.6.

P ROOF. We just established that Picτ (X /R) is a finite and flat group scheme of rank 2 over R, say, isomorphic to Gb,a . By [Ra70, Proposition (6.2.1)], there exists a finite flat Ga,b -torsor π : Xe → X . From the proof of Proposition 4.4 we infer a short exact sequence 0 → OX → π∗ OXe → ωX → 0 .

Now, let L be an extension of L to X . Since h1 (X, L) = 0 by [C-D89, Theorem 1.5.1], global sections of L extend to global sections of L. Clearly, ωX /R ⊗ L extends ωX ⊗ L and h1 (X, ωX ⊗ L) shows that its global sections extend, too. Thus, f∗ L and f∗ (ωX ⊗ L) are free R-modules of rank 3. In particular, π ∗ L defines a map ϕL : Xe 99K P5R that coincides with ϕL on the special fiber. By Proposition 3.7, ϕL is a morphism whose image is the complete intersection of three quadrics and so the same is true for ϕL by openness

20

CHRISTIAN LIEDTKE

of these properties. As in the proof of Proposition 3.7 we conclude that ϕL is Ga,b -equivariant. Moreover, since the three quadrics in the special fiber cutting out e are Ga,b -invariant, the same is true for the quadrics cutting out ϕ (Xe). ϕL (X) L Deformations and lifting. Now, we come to one of our main results. Let X be an Enriques surface with at worst Du Val singularities over k and L be an invertible sheaf on X. We define the functor local Artin algebras Def X,L : → ( Sets ) with residue field k

that associates to each R the set of pairs (X , L), where X is a flat deformation of X over R and L extends L to X . By Proposition 4.4, Picτ (X /R) is a finite and flat group scheme of length 2 over R. We denote by Def Grp,2 the functor that assigns to each local Artin algebra R with residue field k the set of finite flat group schemes of rank 2 over R. Such group schemes are of the form Ga,b for some a, b ∈ R with ab = 2. Thus, Def Grp,2 has W (k)[[a, b]]/(ab − 2) as pro-representable hull. Theorem 4.6. Let X be an Enriques surface with at worst Du Val singularities together with an ample Cossec–Verra polarization L. Then the morphism of functors Def X,L → Def Grp,2 ,

that assigns to each flat deformation X /R its Picτ (X /R), is smooth.

Remark 4.7. We shall see in Section 5 that this fails to be true if L is not ample or if we consider unpolarized deformations. P ROOF. Existence of this morphism follows from Proposition 4.4. Now, let R′ → R be a small extension, let (X , L) be a deformation of (X, L) over R and let G ′ be a finite flat group scheme extending G := Picτ (X /R) to R′ . To prove smoothness, we have to find an extension of (X , L) to R′ whose relative Picτ is isomorphic to G ′ . By Proposition 4.5, there exists a G D -torsor π : Xe → X and π ∗ (L) defines an embedding into P5R , whose image is a complete intersection of three G D -invariant quadrics. From the explicit description in Lemma 3.6, we see that we can find a G ′D -action on P5R′ , as well as a complete intersection of G ′D -invariant quadrics Xe′ , extending Xe together with its G D -action on P5R to R′ . Next, consider the morphism Ψ : P5R′ → P11 R′

given by the G ′D -invariant quadrics of Lemma 3.6. By the same lemma, the G ′D invariants of even degree in R′ [x1 , y1 , ..., x3 , y3 ] are generated by these 12 invariant quadrics. Thus, Ψ is the quotient morphism by G ′D . In particular, X ′ := Ψ(Xe′ ) ∼ = Xe′ /G ′D is flat over Spec R′ , extending Xe/G D ∼ = X to R′ . Finally, the G ′D -action defines a descent data on OP5 ′ (1)|Xe′ . Thus, by finite flat R descent, it comes from an invertible sheaf L′ on X ′ , which extends L. As a direct consequence, we obtain lifting to characteristic zero - we note that the ramification needed for the lifting is entirely controlled by Picτ .

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Theorem 4.8. Let X be an Enriques surface with at worst Du Val singularities. Assume that X admits an ample Cossec–Verra polarization. (1) If X is not supersingular then it lifts over W (k), √ and (2) if X is supersingular then it lifts over W (k)[ 2], but not over W (k). ∼ Ga,b . If X is not P ROOF. We fix a, b ∈ k with ab = 2 such that Picτ (X) = ′ ′ supersingular then a 6= 0 or b 6= 0 and we can find a , b ∈ W (k) with a′ b′ = 2 mapping to a, b, and thus a lift of Ga,b to W (k). Lifting of X over W (k) then follows by running through the proof of Theorem 4.6 with R = k and R′ = W (k). Of course, R′ → R is not a small extension but the proof also works in this case. Alternatively, Theorem 4.6 provides us with a formal lifting of X over Spf W (k), which is algebraizable by Proposition 4.1. √ If X√is supersingular, then a = b = 0 and Gπ,π with π := 2 lifts G0,0√∼ = α2 to W (k)[ 2]. The same arguments as before show that X lifts over W (k)[ 2]. On the other hand, if X were a lifting of X over W (k) then Picτ (X /R) would be a finite flat group scheme of length 2 over W (k) with special fiber α2 by Proposition 4.4. However, this contradicts the fact that αp does not admit liftings over W (k), see also [O-T70, Theorem 2]. Moduli spaces. For an algebraically closed field k of characteristic p ≥ 0, we consider the set MCV,ample of pairs (X, L), where - X is an Enriques surfaces with at worst Du Val singularities over k, and - L is an ample Cossec–Verra polarization on X. We end this section by giving this set the structure of an Artin stack and describing its geometry: Theorem 4.9. MCV,ample is a quasi-separated Artin stack of finite type over k. (1) If p 6= 2 then MCV,ample is irreducible, unirational, 10-dimensional and smooth over k. (2) If p = 2 then MCV,ample consists of two irreducible, unirational, smooth and 10-dimensional components µ2 MCV,ample

and

Z/2Z

MCV,ample .

Moreover, - they intersect transversally along an irreducible, unirational, smooth α2 and 9-dimensional closed substack MCV,ample , α2 - MCV,ample parametrizes supersingular surfaces, α2 G - MCV,ample − MCV,ample parametrizes singular surfaces (G = µ2 ) and classical surfaces (G = Z/2Z), respectively, G contains an open and dense substack, whose - for all G, MCV,ample geometric points correspond to smooth surfaces. P ROOF. We only discuss the case p = 2, since the analysis for p 6= 2 is analogous to the case of singular Enriques surfaces in characteristic 2. First of all, since for every (X, L) ∈ MCV,ample , X is projective and since every formal deformation is effective by Proposition 4.1, the set MCV,ample can

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be given the structure of a quasi-separated Artin stack of finite type over k, see [Ar74a, Example (5.5)] for a sketch, or [Ri96] for a detailed discussion. Next, let G be a finite flat group scheme of length 2 and consider the GD -action on P5 as in Lemma 3.6. Let Ψ : P5 → P11 be the morphism defined by the GD invariant quadrics as in the proof of Theorem 4.6. As explained in [B-M76, §3], the inverse image of a generic hyperplane section yields the canonical double cover of an Enriques surface together with a GD -action. Such surfaces are overparametrized by an open dense subset UG of the Grassmannian Grass(3, 12). It follows from Proposition 3.7, that all Enriques surfaces with at worst Du Val singularities, ample Cossec–Verra polarization and whose canonical double cover is a GD -torsor arise this way. If we denote by N G the substack of surfaces with Picτ ∼ = G, then we G have just shown that UG maps dominantly onto N , showing irreducibility, as well as unirationality. Now, if X → S is a family of Enriques surfaces with at worst Du Val singularities, then the set of points such that Xs is a classical Enriques surface is open, since the property h1 (OX ) = 0 is open by semi-continuity. Also, the set of points s.th. Xs is singular is open: by [Ar74b] there exists a surjective map S ′ → S and an algebraic space X ′ → S ′ that simultaneously resolves the singularities of X → S. But then, the property of being a singular Enriques surface, i.e., satisfying h0 (ΩX ) = 0, is open on S ′ by semi-continuity. Now, for each s ∈ S, the h → Oh map on Henselizations OS,s S ′ ,s′ is finite. We conclude that being a singular Enriques surface is stable under generization also on S, proving openness. Similar arguments show that the set of points such that Xs is supersingular, is closed. We conclude that N G for G = µ2 and for G = Z/2Z belong to different components G of MCV,ample . We denote these components by MCV,ample and remark that they G contain N as open and dense substacks. Next, let (X, L) ∈ N α2 . The group scheme G0,t over k[[t]] has special fiber α2 and generic fiber µ2 . By Theorem 4.6, there exists a family X → Spec k[[t]] with special fiber X and Picτ (X /k[[t]]) ∼ = G0,t . In particular, the geometric generic fiber of this family is a singular Enriques surface. This shows that N α2 is a closed µ2 substack of MCV,ample . Using Gt,0 instead, we conclude that N α2 is also a closed

µ2 and MCV,ample intersect along N α2 . substack of MCV,ample . Thus, MCV,ample The functor Def Grp,2 restricted to the subcategory of local Artin k-algebras has pro-representable hull k[[x, y]]/(xy), see the discussion before Theorem 4.6. But then, the statements about smoothness and transversal intersections follow from Theorem 4.6. Generic hyperplane sections of Ψ(P5 ) ⊆ P11 (notation as in Theorem 4.6) yield smooth singular, classical and supersingular Enriques surfaces with ample Cossec–Verra polarizations, see also [B-M76, §3]. Since ampleness and smoothG for ness are open properties, there exist open and dense substacks of MCV,ample G = µ2 , Z/2Z, and α2 , respectively, corresponding to smooth surfaces. We partly postpone the computation of the dimension to Section 5: namely, for G = Z/2Z and µ2 we shall prove there that the just-established open and G parametrizing smooth surfaces are isomorphic to dense substacks of MCV,ample Z/2Z

Z/2Z

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23

open substacks of the yet to be defined stack MCV,smooth . Thus, it will suffice to compute the dimension at such points of MCV,smooth , which is equal to 10, see the discussion after Proposition 5.3. From this, the dimension and the local description α2 of MCV,ample follows from Theorem 4.6 and the description of Def Grp,2 . Remark 4.10. Over the complex numbers, Casnati [Ca04] considered Enriques surfaces together with invertible sheaves with self-intersection 4, and showed that the corresponding moduli space is rational. Clearly, not every such polarization is Cossec–Verra. Nevertheless, in view of this result it would be interesting to know whether the components of MCV,ample are rational. Clearly, we do not have to restrict ourselves to deformations over k-algebras. We leave it to the reader to use the previous proof to show that in fact Theorem 4.11. The moduli space MCV,ample is a quasi-separated Artin stack of finite type and smooth over Spec Z[x, y]/(xy − 2). 5. U NPOLARIZED

MODULI

In this section we study deformations, moduli and liftings of smooth Enriques surfaces - with and without Cossec–Verra polarizations, leading to the moduli spaces MCV,smooth and MEnriques . The general picture is similar to the one of the previous section. These two moduli spaces are related to MCV,ample via Artin’s functor that simultaneously resolves Du Val singularities in families. At points corresponding to classical and exceptional Enriques surfaces, it is this functor which is responsible for the non-smoothness of MEnriques and MCV,smooth . Let us consider the following sets: (1) MEnriques , whose elements are smooth Enriques surfaces. (2) MCV,ample , whose elements are pairs (X, L), where - X is an Enriques surface with at worst Du Val singularities, and - L is an ample Cossec–Verra polarization. (3) MCV,smooth , whose elements are pairs (X, L), where - X is a smooth Enriques surface, and - L is a Cossec–Verra polarization. (4) MGrp,2 , whose elements are finite flat group schemes of length 2. Clearly, all these sets carry structures of algebraic stacks, but let us defer the discussion of quasi-separatedness until Theorem 5.2 below. In any case, they are related by two functors: MCV,smooth m Φcont mmmm m m mm v mm m

MCV,ample

The first functor contracts nodal curves: Φcont : MCV,smooth → (X, L)

QQQ QQΦ QQforget QQQ QQ(

MEnriques

MCV,ample L 7 → (X ′ := Proj n≥0 H 0 (X, L⊗n ), OX ′ (1))

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Now, a morphism from some base scheme S to MCV,ample corresponds to a family X → S of Enriques surfaces with at worst Du Val singularities together with an ample Cossec–Verra polarization L. Thus, its fiber is simultaneous resolutions of singularities of −1 ′ Φcont (S) : S /S 7→ X ×S S ′ → S ′ together with the pullback of L Artin [Ar74b] showed that this functor is representable by a locally quasi-separated and quasi-finite algebraic space over S. Thus, Proposition 5.1. The functor Φcont is representable, locally quasi-separated and a bijection on geometric points. By Theorem 4.9, every component of MCV,ample contains an open and dense substack over which Φcont is in fact an isomorphism. As an application we obtain quasi-separatedness of our stacks. We note that this is in contrast to K3 surfaces, where the moduli space of unpolarized surfaces is highly non-separated. Theorem 5.2. The moduli spaces MEnriques , MCV,ample and MCV,smooth are quasi-separated Artin stacks of finite type over k. P ROOF. We have to verify that the assumptions of [Ar74a, Theorem (5.3)] are fulfilled. As explained in [Ar74a, Example (5.5)], this is clear for MCV,ample , see also Theorem 4.9. In the remaining cases, everything except quasi-separatedness is clear. But now, quasi-separatedness of MCV,ample together with Proposition 5.1 implies quasi-separatedness of MCV,smooth . Finally, since MCV,smooth → Spec k factors over MEnriques → Speck, we obtain quasi-separatedness of MEnriques . The second functor forgets the Cossec–Verra polarization Φforget : MCV,smooth → MEnriques (X, L)

7→

X

Here, we have the following: Proposition 5.3. The functor Φforget is representable, separated and locally finite and flat. It is smooth at a geometric point (X, L) if and only if X is classical. P ROOF. A family X → S of Enriques surfaces is automatically projective by Proposition 4.1. By [FGA, no 232, Theorem 3.1], Pic(X /S) is representable by a separated scheme, which is locally of finite type over S. The condition L2 = 4 is closed and such invertible sheaves lie discrete in the N´eron–Severi lattice. From Theorem 2.1 and Theorem 2.4 we see that the Φ ≥ 2 is characterized by the property that ϕL is birational. Since this is an open property, so is the condition Φ ≥ 2. On the other hand, invertible sheaves with selfintersection number 4 satisfy Φ ≤ 2 by [C-D89, Lemma 3.6.1], and thus Φ = 2 is an open property for such invertible sheaves. We conclude that Φ−1 forget (S) is τ locally represented by Pic (X /S), proving local finiteness and flatness of Φforget . In particular, Φforget is smooth at (X, L) if and only if Picτ (X) is reduced, which is the case if and only if X is classical.

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25

An invertible sheaf L on a smooth surface X determines via d log a class in H 1 (X, ΩX ), the Chern class of L, which can also be interpreted as an extension class in Ext1 (ΘX , OX ), the Atiyah extension of L 0 → OX → AL → ΘX → 0 .

The groups H i (X, ΘX ) provide a tangent-obstruction theory for MEnriques , and the groups H i (X, AL ) a tangent-obstruction theory for MCV,smooth . Then, the differential of Φforget can be computed from the cohomology sequence dΦforget

... → H 1 (OX ) → H 1 (AL ) −→ H 1 (ΘX ) → H 2 (OX ) → ... If X is classical, then dΦforget is an isomorphism, whereas it has one-dimensional kernel and cokernel in the non-classical case. In particular, we find h0 (AL ) = h0 (ΘX ) + 1,

h1 (AL ) = h1 (ΘX ),

and

h2 (AL ) = h2 (ΘX ).

The hi (ΘX ) are well-known, see Section 1. In particular, if (X, L) is is neither supersingular nor exceptional, MCV,smooth is 10-dimensional at (X, L). Using the previously defined functors, we now compare our moduli spaces: Theorem 5.4. In characteristic 6= 2, all three stacks MEnriques , MCV,smooth and MCV,ample are smooth, irreducible, unirational and 10-dimensional. P ROOF. We have shown this for MCV,ample in Theorem 4.9. There exists an open and dense substack, over which MCV,smooth and MCV,ample are isomorphic, showing irreducibility and unirationality of MCV,smooth . The latter implies that MEnriques is irreducible and unirational. Smoothness of MCV,smooth and MEnriques follows from h2 (ΘX ) = h2 (AL ) = 0. Remark 5.5. Using analytic methods, Kond¯o [Ko94] has shown that MEnriques over the complex numbers is rational. It would be interesting to know whether this is also true in positive characteristic. We described the geometry of MCV,ample in characteristic 2 in Theorem 4.9, and extend this now to the other moduli stacks. Theorem 5.6. Let M be one of MEnriques , MCV,ample and MCV,smooth . Then it consists of two 10-dimensional, irreducible and unirational components M µ2

and

M Z/2Z .

Moreover, - they intersect along a closed substack M α2 , which is 9-dimensional, irreducible and unirational, - M α2 parametrizes supersingular Enriques surfaces, - M G − M α2 parametrizes singular (G = µ2 ) and classical (G = Z/2Z) Enriques surfaces, respectively. The local geometry is a little bit more tricky and given by the following result

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Theorem 5.7. Let X be an Enriques surface in characteristic 2 and L a Cossec– Verra polarization. - If X is singular, or classical and not exceptional, then all three moduli stacks are smooth at X, (X, L), and Φcont (X, L), respectively. - If X is classical and exceptional, then - MCV,ample is smooth at Φcont (X, L), whereas - MCV,smooth and MEnriques are not smooth at (X, L) and X. More precisely, they acquire locally irreducible hypersurface singularities. - If X is supersingular, then the intersection of M µ2 and M Z/2Z in X is transversal - for Φcont (X, L) ∈ MCV,ample , - for (X, L) ∈ MCV,smooth if L is ample, and - for X ∈ MEnriques , if X admits an ample Cossec–Verra polarization. The latter two conditions hold along an open and dense substack of M α2 . Remarks 5.8. Let us note that (1) we do not know whether the intersection at supersingular points is always transversal, and that (2) although generic Enriques surfaces satisfy h0 (ΘX ) = 0, exceptional and supersingular Enriques surfaces fulfill h0 (ΘX ) = 1. This implies that MEnriques cannot be a Deligne–Mumford stack in characteristic 2. P ROOF (of both theorems). The assertions on dimension, irreducibility and unirationality are shown as in the proof of Theorem 5.4. We established all stated properties of MCV,ample in Theorem 4.9. Moreover, smoothness in the singular or classical and non-exceptional case follows as in the proof of Theorem 5.4. If X is a classical and exceptional Enriques surface, then h1 (AL ) − h0 (AL ) = h1 (ΘX ) − h0 (ΘX ) = 11. However, the moduli spaces are 10-dimensional at this point, and so MEnriques and MCV,smooth cannot be smooth at X and (X, L), respectively. Since the obstruction spaces H 2 (ΘX ) and H 2 (AL ) are 1-dimensional, the singularities are hypersurface singularities. As Φcont is a bijection on geometric points, and MCV,ample is smooth, we conclude local irreducibility. Now, let X be supersingular. Transversal intersection of two components at Φcont (X, L) ∈ MCV,ample has been established in Theorem 4.9. Using Φcont and Φforget , we conclude that also the other two moduli spaces consist of two components intersecting along the supersingular locus. Now, if L is ample then MCV,smooth and MCV,ample are locally isomorphic near (X, L), and thus the intersection is transversal. Let us now prove transversally at X ∈ MEnriques in case X admits an ample Cossec–Verra polarization L: then (X, L) is a geometric point of MCV,ample . By Theorem 4.6, MCV,ample → MGrp,2 is smooth and in particular, every first-order deformation of α2 can be extended to a first-order deformation of the pair (X, L). Forgetting L, we infer that the differential of the map MEnriques → MGrp,2 at X is surjective. Given a deformation X → S of S, and a small extension S → S ′ such that Picτ (X /S) cannot be extended to S ′ , also X → S cannot be extended to S ′ by Proposition 4.4. Thus, the map of obstruction spaces ob(MEnriques ) → ob(MGrp,2 ) is non-trivial. Both spaces

ENRIQUES SURFACES

27

are 1-dimensional, and so the map is in particular injective. This implies that MEnriques → MGrp,2 is in fact smooth at X. Again, we conclude that the intersection of the components at X is transversal. Remark 5.9. At a point of MEnriques corresponding to a singular and exceptional Enriques surface X, the following happens: There exists a family X → S over some local base with special fiber X, as well as a small extension S → S such that the family cannot be extended over S. After choosing a Cossec–Verra polarization L on X, this polarization extends uniquely to X (since X is classical), and Φcont yields a family X ′ → S. Since MCV,ample ′ is smooth at Φcont (X, L), the family X ′ → S extends to a family X → S. By construction, X → S is a simultaneous resolution of singularities of X ′ → S over ′ S. By assumption, a simultaneous resolution of singularities of X → S extending X → S does not exist over S. However, it does exist after a ramified extension of S by Artin’s result [Ar74b]. Summing up, the singularities of MEnriques at X can be explained via ADEcurves and obstructions coming from Artin’s simultaneous resolution functor. Over the complex numbers, similar phenomena have been described in [B-W74]. Let us finally address the lifting problem for Enriques surfaces. Every Enriques surface X admits a Cossec–Verra polarization L. After contracting those nodal curves that have zero-intersection with L, we obtain an Enriques surface X ′ with at worst Du Val singularities. In Theorem 4.8 we have shown √ that X ′ lifts over W (k) unless it is supersingular, in which case it lifts over W (k)[ 2]. Here is what we know about lifting of smooth Enriques surfaces: Theorem 5.10. Let X be an Enriques surface in positive characteristic p. (1) If p 6= 2 then X lifts over W (k). (2) If X is a singular, or a classical and non-exceptional Enriques surface, then it lifts over W (k). (3) If X is supersingular √ and admits an ample Cossec–Verra polarization then it lifts over W (k)[ 2], but not over W (k). (4) In the remaining cases, X lifts over a possibly ramified extension of W (k). P ROOF. In the first two cases, we have h2 (ΘX ) = 0, i.e., there exists a formal lifting over √ W (k), whose algebraization follows from Proposition 4.1. Lifting over W (k)[ 2] and non-lifting over W (k) for supersingular surfaces admitting ample Cossec–Verra polarizations has been established in Theorem 4.8. In the remaining cases, Theorem 5.6 tells us that the deformation functor has only locally irreducible hypersurface singularities, i.e., there exists a formal lifting over a ramified extension of W (k), which is algebraizable by Proposition 4.1. Remark 5.11. The first two results are well-known, see [La83] and [E-SB04]. It would be interesting to know whether exceptional classical √ surfaces lift over W (k) and, whether all supersingular surfaces lift over W (k)[ 2].

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R EFERENCES [ACGH]

E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of algebraic curves. Vol. I., Grundlehren der Mathematischen Wissenschaften 267, Springer 1985. [Ar74a] M. Artin, Versal Deformations and Algebraic Stacks, Invent. Math. 27, 165-189 (1974). [Ar74b] M. Artin, Algebraic construction of Brieskorn’s resolutions, J. Algebra 29, 330-348 (1974). [B-P83] W. Barth, C. Peters, Automorphisms of Enriques surfaces, Invent. Math. 73, 383-411 (1983). [B-M76] E. Bombieri, D. Mumford, Enriques classification of surfaces in char.p, III, Invent. Math. 35, 197-232 (1976). [B-W74] D. M. Burns, J. M. Wahl, Local contributions to global deformations of surfaces, Invent. Math. 26, 67-88 (1974). [Ca04] G. Casnati, The moduli space of Enriques surfaces with a polarization of degree 4 is rational, Geom. Dedicata 106, 185-194 (2004). [CFHR] F. Catanese, M. Franciosi, K. Hulek, M. Reid, Embeddings of curves and surfaces, Nagoya Math. J. 154, 185-220 (1999). [Co83] F. R. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280, 737-751 (1983). [Co85] F. R. Cossec, Projective Models of Enriques Surfaces, Math. Ann. 265, 283-334 (1983). [C-D85] F. R. Cossec, I. Dolgachev, On automorphisms of nodal Enriques surfaces, Bull. Amer. Math. Soc. 12, 247-249 (1985). [C-D89] F. R. Cossec, I. Dolgachev, Enriques surfaces I, Progress in Mathematics 76, Birkh¨auser 1989. [Do84] I. Dolgachev, On automorphisms of Enriques surfaces, Invent. Math. 76, 163-177 (1984). [E-H87] D. Eisenbud, J. Harris, On Varieties of Minimal Degree (A Centennial Account), Algebraic Geometry, Bowdoin 1985, Proc. Symp. Pure Math. 46, Part 1, 3-13 (1987). [E-S?] T. Ekedahl, N. I. Shepherd-Barron, unpublished results on Enriques surfaces, discussed by Ekedahl and Partsch on mathoverflow.net on April 11-12, 2010. [E-SB04] T. Ekedahl, N. I. Shepherd-Barron, On exceptional Enriques surfaces, arXiv:math/0405510 (2004). [En08] F. Enriques, Un’ osservazione relativa alla superficie di bigenere uno, Bologna Rend. 12, 40-45 (1908). [FGA] A. Grothendieck, Fondements de la g´eom´etrie alg´ebrique, Extraits du S´eminaire Bourbaki, 1957–1962. ´ [Il79] L. Illusie, Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. Ecole Norm. Sup. 12, 501-661 (1979). [Jo83] J.-P. Jouanolou, Th´eor`emes de Bertini et applications, Progress in Mathematics 42, Birkh¨auser 1983. [Ko94] S. Kond¯o, The rationality of the moduli space of Enriques surfaces, Compositio Math. 91, 159-173 (1994). [La83] W. E. Lang, On Enriques surfaces in characteristic p. I, Math. Ann. 265, 45-65 (1983). ´ [O-T70] F. Oort, J. Tate, Group schemes of prime order, Ann. Sci. Ecole Norm. Sup. 3, 1-21 (1970). ´ [Ra70] M. Raynaud, Sp´ecialisation du foncteur de Picard, Inst. Hautes Etudes Sci. Publ. Math. 38, 27-76 (1970). [Ri96] J. Rizov, Moduli stacks of polarized K3 surfaces in mixed characteristic, Serdica Math. J. 32, 131-178 (2006). [SD74] B. Saint-Donat, Projective models of K3 surfaces, Amer. J. Math. 96, 602-639 (1974). [Sa03] P. Salomonsson, Equations for some very special Enriques surfaces in characteristic two, arxiv:math.AG/0309210 (2003). [Sch91] F.-O. Schreyer, A standard basis approach to syzygies of canonical curves, J. Reine Angew. Math. 421, 83-123 (1991).

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[Ve83]

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A. Verra, The e´ tale double covering of an Enriques’ surface, Rend. Sem. Mat. Univ. Politec. Torino 41, 131-167 (1983).

D EPARTMENT OF M ATHEMATICS , S TANFORD U NIVERSITY, 450 S ERRA M ALL , S TANFORD CA 94305-2125, USA E-mail address: [email protected]

arXiv:1007.0787v1 [math.AG] 6 Jul 2010

CHRISTIAN LIEDTKE July 4, 2010

A BSTRACT. We construct the moduli space of Enriques surfaces in positive characteristic and determine its local and global structure. Also, we prove that Enriques surfaces lift to characteristic zero. The key ingredient is that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in P5 , even in characteristic 2.

I NTRODUCTION In order to give examples of algebraic surfaces with h1 (OX ) = h2 (OX ) = 0 that are not rational, Castelnuovo and Enriques constructed the first Enriques surfaces at the end of the 19th century. From the point of view of the Kodaira– Enriques classification, these surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. More precisely, these four classes consist of Abelian surfaces, K3 surfaces, Enriques surfaces and (Quasi-)Hyperelliptic surfaces. In characteristic 6= 2, Enriques surfaces behave extremely nice: deformations are unobstructed by results of Illusie [Il79] and Lang [La83]. Over the complex numbers, their moduli space is irreducible, smooth, unirational and 10-dimensional, and Kond¯o [Ko94] showed even rationality. Next, their fundamental groups are of order 2 and their universal covers are K3 surfaces. Moreover, Cossec [Co85] and Verra [Ve83] found explicit equations of these K3-covers: they are birational to complete intersections of three quadrics in P5 and the Z/2Z-action can be written down explicitly. (For generic Enriques surfaces this was already known to Enriques himself [En08].) Finally, Enriques surfaces in characteristic 6= 2 lift over the Witt ring, which is due to Lang [La83]. In characteristic 2, the situation is more complicated: first of all, as shown by Bombieri and Mumford [B-M76], Enriques surfaces fall into three classes, called classical, singular and supersingular. Although they still possess canonically defined flat double covers, which have trivial dualizing sheaves and which ”look“ cohomologically like K3 surfaces, these are in general only integral Gorenstein surfaces and may not be normal. It also happens that deformations are obstructed and finally, supersingular Enriques surfaces do not lift over the Witt ring. In this article, we clarify the situation in positive characteristic, and especially in characteristic 2. 2000 Mathematics Subject Classification. 14J28, 14J10. 1

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CHRISTIAN LIEDTKE

We start with the description of K3(-like) covers of Enriques surfaces. e → X be the K3(-like) cover of an Enriques surface X. Then Theorem. Let π : X there exists a morphism ϕ e → ϕ(X) e ⊆ P5 X e is a complete intersection of that is birational onto its image. The image ϕ(X) three quadrics. The exceptional locus of ϕ is, in a certain sense, a union of ADE-curves, see Theorem 3.1. Next, π is a torsor under a finite flat group scheme G of length 2. We describe the linear G-action on P5 induced by ϕ, and equations of the G-invariant e in Proposition 3.7. As a byproduct, we obtain quadrics cutting out ϕ(X)

Corollary. All Enriques surfaces in arbitrary characteristic arise via the Bombieri– Mumford–Reid construction in [B-M76, §3]. This result is the key to determining the local and global structure of the moduli space MEnriques of Enriques surfaces in positive characteristic p: Theorem. MEnriques is a quasi-separated Artin stack of finite type over k. (1) If p 6= 2, then MEnriques is irreducible, unirational, 10-dimensional and smooth over k. (2) If p = 2, then MEnriques consists of two irreducible, unirational and 10dimensional components Z/2Z

MEnriques

and

µ2 MEnriques .

Moreover, - they intersect along an irreducible, unirational and 9-dimensional α2 closed substack MEnriques , α2 - MEnriques parametrizes supersingular Enriques surfaces, µ2 - MEnriques − MEnriques is smooth and parametrizes singular surfaces, Z/2Z

Z/2Z

Z/2Z

- MEnriques − MEnriques parametrizes classical Enriques surfaces. It is smooth outside the locus of exceptional Enriques surfaces. Exceptional Enriques surfaces were introduced by Ekedahl and Shepherd-Barron in [E-SB04]. We note that Bombieri and Mumford conjectured, or, at least hoped for such a general picture already back in [B-M76] - except for the appearance of exceptional Enriques surfaces, on which we will comment below. To obtain these results, we first study polarized moduli, which turn out to be interesting in their own right, as they behave much nicer. In a certain sense, there is a class of natural and minimal polarizations for our setup, cf. the discussion at the beginning of Section 3, which is the following: Definition. A Cossec–Verra polarization on an Enriques surface X is an invertible sheaf L with self-intersection number 4 and such that every genus-one fibration |2E| on X satisfies L · E ≥ 2.

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In general, such invertible sheaves are not ample, but only big and nef. However, it is important to note that every Enriques surface possesses such a polarization. This is different from algebraic K3 surfaces, which are all polarizable, but where we need infinitely many types of polarizations to capture every one of them. Although every Enriques surface X possesses a Cossec–Verra polarization L, it is in general not unique. We refer to Proposition 3.4 for quantitative results. Contracting those curves that have zero-intersection with L, we obtain a pair (X ′ , L′ ), where X ′ is an Enriques surface with at worst Du Val singularities and L′ is an ample Cossec–Verra polarization. Theorem. Let X ′ be an Enriques surface over k with at worst Du Val singularities admitting an ample Cossec–Verra polarization. (1) If X ′ is not supersingular then it lifts over the√ Witt ring W (k). ′ (2) If X is supersingular then it lifts over W (k)[ 2], but not over W (k). Next, we denote by MCV,ample the moduli space of pairs (X, L), where X is an Enriques surface with at worst Du Val singularities and L is an ample Cossec–Verra polarization. This moduli space behaves extremely nice, even in characteristic 2: Theorem. MCV,ample is a quasi-separated Artin stack of finite type over k. (1) If p 6= 2 then MCV,ample is irreducible, unirational, 10-dimensional and smooth over k. (2) If p = 2 then MCV,ample consists of two irreducible, unirational, smooth and 10-dimensional components µ2 MCV,ample

and

Z/2Z

MCV,ample .

Moreover, - they intersect transversally along an irreducible, unirational, smooth α2 and 9-dimensional closed substack MCV,ample , α2 - MCV,ample parametrizes supersingular surfaces, α2 G − MCV,ample parametrizes singular surfaces (G = µ2 ) - MCV,ample and classical surfaces (G = Z/2Z), respectively, The lifting result and the description of MCV,ample give a beautiful picture of how Enriques surfaces ”should“ behave. However, we already mentioned above that MEnriques is not smooth along at points corresponding to classical and exceptional Enriques surfaces. Here is the reason: MCV,ample and MEnriques are related via simultaneous resolutions of singularities and then forgetting the polarization. It is Artin’s simultaneous resolution functor [Ar74b] which is responsible for singularities of MEnriques . This is similar to canonically polarized surfaces of general type, where Burns and Wahl [B-W74] showed how singularities of the canonical models may obstruct moduli spaces. We refer to Remark 5.9 for details. Because of these difficulties, we lose control over the ramification needed in order to lift Enriques surfaces. However, it suffices to prove the following: Theorem. Enriques surfaces lift to characteristic zero.

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We refer to Theorem 5.10 for what we know about ramification. Lifting over W (k) in characteristic p 6= 2, as well as for singular Enriques surfaces in p = 2 has been established by Lang [La83]. This article is organized as follows: After reviewing a couple of general facts in Section 1, we study projective and birational models of the canonical double cover of Enriques surfaces in Section 2. These results extend previous work of Cossec [Co85] to characteristic 2. The main difficulty is that Saint-Donat’s results [SD74] on linear systems on K3 surfaces cannot be applied to this double cover and we have to find rather painful ways around. In Section 3 we show that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in P5 . We introduce the notion of a Cossec–Verra polarization and establish a couple of general facts about them. Finally, we explicitly describe the action of the finite flat group scheme of length two, which acts on this complete intersection. In particular, we will see that every Enriques surface arises via the Bombieri–Mumford–Reid construction of [B-M76]. In Section 4 we study pairs of Enriques surfaces with at worst Du Val singularities together with ample Cossec–Verra polarizations. We prove their lifting to characteristic zero and construct the moduli space MCV,ample of such pairs. Using the results of Section 3, all boils down to describing deformations and moduli of complete intersections of three quadrics together with the action of a finite flat group scheme of length 2. Finally, in Section 5 we relate MCV,ample to the moduli space MEnriques of unpolarized Enriques surfaces. These are connected via Artin’s functor of simultaneous resolutions of singularities and the functor that forgets the Cossec–Verra polarization. Finally, we prove lifting to characteristic zero. Remark. While working on this article and discussing it, I was pointed out the discussion [E-S?] on the internet platform mathoverflow.net. It seems that some of the results in Section 5 have been obtained independently by Ekedahl and Shepherd-Barron several years ago, but were never published. Acknowledgements. I thank Brian Conrad, Igor Dolgachev, David Eisenbud and Jack Hall for long discussions and comments. Moreover, I gratefully acknowledge funding from DFG under research grant LI 1906/1-1 and thank the department of mathematics at Stanford university for kind hospitality. 1. G ENERALITIES We start by recalling a couple of general facts on Enriques surfaces, and refer to [B-M76] and [C-D89] for details and further references. Throughout this article, X denotes an Enriques surface over an algebraically closed field k of arbitrary characteristic p ≥ 0. By the definition of Bombieri and Mumford, this means ωX ≡ OX

and

b2 (X) = 10,

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where ≡ denotes numerical equivalence. Moreover, we have χ(OX ) = 1

In characteristic p 6= 2 we inequality h1 (OX ) ≤ 1 is true. sense to study the action of the either zero or a bijection.

and

b1 (X) = 0 .

have h1 (OX ) = 0, whereas for p = 2 only the Thus, if h1 (OX ) happens to be non-zero, it makes absolute Frobenius F on H 1 (OX ), which must be

Definition 1.1. An Enriques surface X is called ⊗2 ∼ (1) classical if h1 (OX ) = 0, hence ωX ∼ 6 OX and ωX = OX , = (2) singular if h1 (OX ) = 1, hence ωX ∼ = OX and F is bijective on H 1 (OX ), (3) supersingular if h1 (OX ) = 1, hence ωX ∼ = OX and F is zero on H 1 (OX ). The Picard scheme of X is smooth only if it is classical. More precisely, Picτ (X) is isomorphic to Z/2Z (classical), µ2 (singular) or α2 (supersingular), respectively. In each case, Picτ gives rise to finite and flat morphism of degree 2 e → X, π : X

which is a torsor under G := (Picτ (X))D , where −D = Hom(−, Gm ) denotes Cartier duality. In particular, if p 6= 2 or if X is a singular Enriques surface then G ∼ = Z/2Z, e the morphism π is an e´ tale Galois cover and X is a K3 surface. In the remaining e is never smooth, possibly even non-normal. cases, π is purely inseparable and X e In any case, X is an integral Gorenstein surface with invariants ωe ∼ = O e , χ(O e ) = 2 and h1 (O e ) = 0, X

X

X

X

i.e., ”K3-like“. Having only an integral Gorenstein surface rather than a smooth K3 surface as double cover, is one of the main reasons why Enriques surfaces in characteristic 2 are so difficult to come by. Finally, let us recall some of the Hodge invariants, see [La83, Theorem 0.11]: p 2

type classical singular supersingular

6= 2

h01 h10 h0 (ΘX ) h1 (ΘX ) h2 (ΘX ) 0 1 a 10 + 2a a 1 0 0 10 0 1 1 1 12 1 0 0 0 10 0

Classical Enriques surfaces satisfy a ≤ 1 and surfaces with a = 1 have been described and explicitly classified by Ekedahl and Shepherd-Barron [E-SB04] and Salomonsson [Sa03]. 2. P ROJECTIVE

MODELS OF THE DOUBLE COVER

In this section we study linear systems and projective models of the K3(-like) e of an Enriques surface X. Since there is no canonical polarization on X, e cover X the best thing to do is to consider pull-backs of invertible sheaves from X with positive self-intersection number. In characteristic 6= 2, this has been carried out by Cossec [Co85, Section 8]: such a pull-back defines a morphism that is either

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birational onto its image or generically finite of degree 2 onto a rational surface. Cossec’s proof relies on Saint-Donat’s analysis [SD74] of linear systems on K3 e In characteristic 2, the main difficulty is that X e is in surfaces that he applies to X. general only an integral Gorenstein surface, and so we have to take rather painful detours. The canonical double cover. We start by describing the canonical double cover of an Enriques surface X over a field k. Since Picτ (X) is a finite and flat group scheme of length 2, it gives rise to a torsor e → X π : X

under (Picτ (X))D , where −D denotes Cartier duality [Ra70, Proposition (6.2.1)]. More precisely, let P be the Poincar´e invertible sheaf on X × Picτ (X). By its universal property, there exists a morphism ψ : Picτ (X) → Picτ (X) such that P⊗P ∼ = (id × ψ)∗ P. Clearly, ψ = µ ◦ ∆, where ∆ is the diagonal and µ is the multiplication map of Picτ (X). Dualizing, we obtain an OX -algebra structure on P ∨ . Dualizing the multiplication map P ⊗ (OX ⊗ OPicτ (X) ) → P, we obtain a OX ⊗ OPicτ (X)D -comodule structure on P ∨ . Putting all this together, we obtain e ∼ π : X = Spec P ∨ → X

together with its (Picτ (X))D -action. In particular, this group scheme acts via its regular representation. Let us recall from [O-T70, Theorem 2] that a finite flat group scheme of length 2 over k is isomorphic to Ga,b for some a, b ∈ k with ab = 2. The assignment ρreg : Ga,b (S) = {s ∈ S | s2 = as} → GL2 (S) 1 s s 7→ 0 1 − bs

for any k-algebra S, defines the regular representation, see also [B-M76, §3] and Lemma 3.6 below. Projective models. As in [C-D89, Chapter III §2], we define Φ for an effective divisor C on X to be 1 Φ(C) := inf E · C, where |E| is a genus one pencil on X 2 For example, if C is an irreducible curve with C 2 > 0 then the linear system |C| is basepoint-free if and only if Φ(C) ≥ 2, see [C-D89, Theorem 4.4.1]. As we shall see now and in Theorem 2.4 below, Φ also controls the behavior of linear e systems on X. Theorem 2.1. Let C be an irreducible curve with C 2 > 0 and Φ(C) ≥ 2 on an Enriques surface X. Then (1) C 2 ≥ 4, e is globally generated, (2) the invertible sheaf π ∗ OX (C) on X ∗ (3) a generic Cartier divisor in |π OX (C)| is an integral Gorenstein curve, which is not hyperelliptic,

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(4) |π ∗ OX (C)| gives rise to a morphism

e → P(1+C 2 ) , ϕ : X

which is birational onto an integral surface of degree 2C 2 . P ROOF. By [C-D89, Theorem 4.4.1] the linear system |C| is base-point free. From the formula for h0 of [C-D89, Corollary 1.5.1] it follows that a generic divisor in |C| is reduced. Since C is irreducible by assumption, a generic divisor in |C| is reduced and irreducible. In particular, |C| has no fixed component. Now, if we had C 2 = 2 then |C| would define a morphism onto P1 , which contradicts C 2 6= 0 and we conclude C 2 ≥ 4. Moreover, since OX (C) is globally generated, e is also globally generated. it follows that π ∗ OX (C) on X Next, we consider the short exact sequence (1)

0 → OX (C) → π∗ π ∗ OX (C) → ωX (C) → 0 .

We have h1 (OX (C)) = 0 by [C-D89, Theorem 1.5.1], which, together with e π ∗ OX (C)) = 2 + C 2 . Thus, π ∗ OX (C) [C-D89, Corollary 1.5.1] implies h0 (X, e to (1+C 2 )-dimensional projective space. Also, gives rise to a morphism ϕ from X e is an since the image of |C| is a surface, the same is true for ϕ. Moreover, ϕ(X) e integral surface, i.e., reduced and irreducible, since X is. e is a smooth K3 surface If p 6= 2 or if X is a singular Enriques surface then X ∗ 2 2 ∗ and we compute (π C) = 2C . Since π OX (C) is globally generated, we find e Since non-degenerate and integral surfaces in PN have 2C 2 = deg ϕ · deg ϕ(X). degree at least N − 1, we conclude deg ϕ ≤ 2. If p = 2 and X is classical or supersingular, then π is a torsor under µ2 or α2 . In particular, π is inseparable and there exists a diagram (2)

X (1/2) 8J

88JJJ 88 J̟J 88 JJJ J% 88 8 e X F 88 88 88 π 8

ϕ

/

P1+C

2

X where F : X (1/2) → X denotes the k-linear Frobenius morphism. The composition ϕ ◦ ̟ corresponds to a linear subsystem of |2C| (here, we identify X with X (1/2) ). Both, ϕ and ̟ are morphisms, we have 2 deg ϕ = deg(ϕ ◦ ̟), as well as (2C)2 = 4C 2 . As before, we find deg ϕ ≤ 2, this time by arguing on X (1/2) . In order to show deg ϕ = 1 (again, for arbitrary π and p), we argue by contradiction similar to the proof of [Co85, Lemma 4.4.3]. So suppose deg ϕ 6= 1. e is an integral surface of degree C 2 in P1+C 2 , Then deg ϕ = 2 and the image ϕ(X) i.e., a surface of minimal degree. These surfaces have been explicitly classified by del Pezzo, see [E-H87] for a characteristic-free discussion. Now, the morphism π is a torsor under a finite flat group scheme G, which is e by G is isomorphic to X and not of length 2 over k. Since the quotient of X

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e it follows that the G-action on X e induces a non-trivial Gisomorphic to ϕ(X) e π ∗ OX (C))) and ϕ(X). e As already seen above, we may write action on P(H 0 (X, ∗ the global sections of π OX (C) as pr

e π ∗ OX (C)) → H 0 (X, ωX (C)) ω e → 0 . 0 → H 0 (X, OX (C)) → H 0 (X, X

Considered as a short exact sequence of G-modules, the G-action is trivial on H 0 (X, OX (C)), as well as H 0 (X, ωX (C)), but possibly not on ωXe . From the discussion at the beginning of this section it follows that H 0 (X, π∗ π ∗ L), as a Gmodule, decomposes into the direct sum of three regular representations of G, see also Proposition 3.7 below. We set P+ := P(H 0 (X, OX (C))), which is the projectivized +id-eigenspace for the G-action. If p 6= 2 or p = 2 and G ∼ = µ2 then the G-action has a second eigenspace, which we may identify with H 0 (X, ωX (C))ωXe . We denote by P− e π ∗ OX (C))) is fixed under the its projectivization. Clearly, if a point in P(H 0 (X, G-action (in the scheme-theoretic sense) then it lies in P+ or P− . For every v ∈ H 0 (X, ωX (C))ωXe , the hyperplane Pv := P(pr−1 (v)) is Ge in an irreducible curve ∆. stable and contains P+ . The generic Pv intersects ϕ(X) 2 2 C Since ∆ is of degree C − 1 in a P , it is a rational normal curve and in particular smooth and rational. Since ∆ is isomorphic to P1 and equipped with a non-trivial G-action, its fixed point scheme has length 2, which is supported in two distinct points if p 6= 2. e contains points that are fixed under G and so its intersection with Thus, ϕ(X) P+ or P− is non-empty. On the other hand, \ e = e P+ ∩ ϕ(X) {s = 0} ∩ ϕ(X) s∈π ∗ H 0 (X,OX (C))∨

e and s ∈ π ∗ H 0 (X, ωX (C))∨ . This implies that and similarly for P− ∩ ϕ(X) OX (C) or ωX (C) is not globally generated, a contradiction. This establishes deg ϕ = 1. By [Jo83, Th´eor`eme I.6.10], a generic Cartier divisor in |π ∗ OX (C)| is irree where ϕ ducible. The same theorem, applied to the open and dense subset of X e is an isomorphism and where X is smooth, shows that a generic Cartier divisor is generically reduced. Now, Cartier divisors on Gorenstein schemes are Gorenstein and in particular Cohen–Macaulay. Thus, a generic Cartier divisor in |π ∗ OX (C)| is irreducible, generically reduced and Cohen–Macaulay, i.e., irreducible and reduced, i.e., integral. e O e ) = 0, we conclude that ϕ induces Using the adjunction formula and H 1 (X, X on D the morphism associated to ωD . Thus, ϕ being birational, the generic D is not hyperelliptic. Hyperelliptic in the non-smooth case simply means that there exists a morphism of degree 2 onto P1 , see [Sch91]. Using Saint-Donat’s analysis [SD74] of linear systems on K3 surfaces, Cossec e in characteristic 6= 2 is cut out by quadrics: [Co85] has shown that ϕ(X)

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e Theorem 2.2 (Cossec+ε). Under the assumptions of Theorem 2.1, the image ϕ(X) 2 in P1+C is projectively normal and cut out by quadrics, whenever (1) char(k) 6= 2, or (2) char(k) = 2 and X is a singular Enriques surface. P ROOF. In characteristic 6= 2, this is shown in [Co85, Section 8]. e is smooth, ϕ is If X is a singular Enriques surface in characteristic 2, then X e has only isolated singularities. But then, a generic divisor birational and ϕ(X) ∗ D ∈ |π OX (C)| is smooth and the whole analysis in [SD74, Section 7] remains valid also in characteristic 2. Thus, [SD74, Theorem 7.2] and [Co85, Lemma 8.1.2] e is projectively normal and cut out by quadrics. show that ϕ(X) e is always cut out by quadrics – this would follow from It is plausible that ϕ(X) e (we refer to [CFHR, Section 3] for numerical 4-connectedness of π ∗ OX (C) on X a discussion of this notion for singular varieties). However, we have only been able to establish this in the special case, in which we are interested in later on: Proposition 2.3. In addition to the assumptions of Theorem 2.1, assume that C 2 = e ⊂ P5 is projectively normal and cut out by quadrics. 4 and Φ(C) = 2. Then ϕ(X)

e is P ROOF. We have to show that the graded ring associated to π ∗ OX (C) on X generated in degree 1 with relations in degree 2 only, i.e., a Koszul algebra. By Theorem 2.1, a generic Cartier divisor D ∈ |π ∗ OX (C)| is an integral and non-hyperelliptic Gorenstein curve of arithmetic genus pa (D) = 5. For n ≥ 1, we e consider the following short exact sequences on X: ⊗n → 0. 0 → π ∗ OX ((n − 1)C) → π ∗ OX (nC) → ωD

Pushing π ∗ OX ((n − 1)C) forward to X and using [C-D89, Theorem 1.5.1], we e π ∗ OX ((n − 1)C)) = 0 for n ≥ 1. Thus, as explained in the proof conclude h1 (X, of part (ii) of [SD74, Theorem 6.1], to prove our assertion, it suffices to show that the canonical ring of D is a Koszul algebra. Before proceeding, we study genus one half-pencils on X more closely: since Φ(C) = 2, there exists a genus one pencil E on X with C · E = 4. Moreover, let E ′ be a genus one curve with |2E ′ | = |E|, i.e., a half-pencil. We now claim: (1) π ∗ OX (E ′ ) is globally generated with h0 = 2 and h1 = 0, and (2) π ∗ OX (C − E ′ ) also satisfies h0 = 2 and h1 = 0. It is globally generated outside π −1 (R), where R is a (possibly empty) union of ADE-curves. We only deal with the case that π is inseparable in characteristic 2 and leave the remaining and easier cases to the reader: the assertions about h0 and h1 of π ∗ OX (E ′ ) follow from pushing it down to X and then using h0 (X, OX (E ′ )) = h0 (X, ωX (E ′ )) = 1, as well as h1 (X, OX (E ′ )) = h1 (X, ωX (E ′ )) = 0. Now, we use diagram (2): the linear system ̟ ∗ |π ∗ OX (E ′ )| is a linear subsystem of |̟ ∗ π ∗ OX (E ′ )| = |F ∗ E ′ | = |2E ′ | = |E|. Since both satisfy h0 = 2, they are equal and so π ∗ OX (E ′ ) is globally generated since OX (E) is. Let us adjust the proof of [Co85, Theorem 5.3.6] to our situation: by Riemann-Roch there exists an effective divisor D such that E ′ + D ∈ |C|, E ′ D = 2, and D 2 = 0. Moreover,

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there exists a divisor E ′′ of canonical type such that D = E ′′ + R with R ≥ 0. Since Φ(C) = 2 we have CE ′′ ≥ 2 and if equality holds then E ′′ is a genus one half-pencil. Thus, 4 = C 2 ≥ CE ′ + CE ′′ ≥ 4, and we conclude CE ′′ = 2 and CR = 0. In particular, if non-empty, R is a union of ADE-curves. The remaining assertions now follow as before, establishing our two claims. Next, let us show that ϕ(D) possesses a simple (pa (D) − 2)-secant: first, we e π ∗ OX (C −E ′ )) = choose a generic Cartier divisor G ∈ |π ∗ OX (E ′ )|. Since h1 (X, 0 ∗ 0 e π OX (C)) surjects onto H (G, π ∗ OX (C)|G ). Using 0, we conclude that H (X, deg π ∗ OX (C)|G = 4 we see that ϕ embeds G as a quartic into some P3 , which is easily seen to be the complete intersection of two quadrics. Thus, a generic hyperplane H of P5 intersects this complete intersection in 4 points in uniform position. e an integral curve ϕ(D), where D ∈ |π ∗ OX (C)|, having This H cuts out on ϕ(X) the stated simple (pa (D) − 2)-secant. Having established this (pa (D) − 2)-secant, [Sch91, Theorem 1.2] and [Sch91, Corollary 1.3] show that the canonical ring of D is generated in degree 1 and has relations in degree ≤ 3. Our proposition is proved once we show that no relations in degree 3 are needed. Suppose that relations in degree 3 are needed. Then, ϕ(D) is contained in an irreducible surface S of degree pa (D)−2 by [Sch91, Theorem 3.1]. By the classification of surfaces of minimal degree [E-H87] together with pa (D) = 5 we find that S is ruled. Moreover, a generic ruling of S intersects ϕ(D) in three distinct smooth points. Thus, D possesses a globally generated invertible sheaf M of degree 3 with h0 = 2 (a “g31 ”) and D is trigonal. Now, we consider L := π ∗ OX (E ′ )|D . Then π ∗ OX (C − E ′ )|D ∼ = ωD ⊗ L∨ and L and ωD ⊗ L∨ are invertible sheaves of degree 4 = 21 deg ωD . Taking cohomology in 0 → π ∗ OX (E ′ − C) → π ∗ OX (E ′ ) → L → 0, e π ∗ OX (E ′ )) and H 0 (X, e π ∗ OX (C − we obtain h0 (D, L) ≥ 2. Moreover, H 0 (X, ′ 0 0 ∨ E )) inject into H (D, L) and H (D, ωD ⊗ L ), respectively. In particular, L is globally generated since π ∗ OX (E ′ ) is. Choosing D generically, we may assume that D does not intersect π −1 (R) and then ωD ⊗ L∨ is globally generated. Since h1 (D, L) 6= 0, Clifford’s inequality implies h0 (D, L) ≤ 3 and equality could only happen if D were hyperelliptic. Thus, h0 (D, L) = h0 (D, ωD ⊗ L∨ ) = 2. This is enough to show that D is not trigonal: by [ACGH, Excercise III.B-5], which works for integral Gorenstein curves in arbitrary characteristic, the invertible sheaf M making D trigonal would have to be a subsheaf of L or ωD ⊗ L∨ , which is absurd. e arising from curves We complete the picture by discussing linear systems on X C on X with Φ = 1: Theorem 2.4. Let C be an irreducible curve with C 2 > 0 and Φ(C) = 1 on an Enriques surface X. Then e is globally generated, (1) the invertible sheaf π ∗ OX (C) on X

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(2) |π ∗ OX (C)| gives rise to a morphism e → P(1+C 2 ) , ϕ : X

which is generically of degree 2 onto a surface of minimal degree C 2 , e is cut out by quadrics. (3) the image ϕ(X)

e π ∗ OX (C)) = 2 + C 2 . P ROOF. As in the proof of Theorem 2.1 we find h0 (X, ∗ Again, |C| has no fixed component and so π OX (C) is globally generated outside a finite (possibly empty) set of points. e is a surface since the image of Let us first assume C 2 ≥ 4. In this case, ϕ(X) the rational map associated to |C| is a surface [C-D89, Theorem 4.5.1]. Seeking a contradiction, we assume that ϕ is birational. As in the proof of Theorem 2.1, we conclude that a generic Cartier divisor D ∈ |π ∗ OX (C)| is an integral Gorenstein curve. Since Φ(C) = 1, there exists a genus one half-pencil E ′ on X such that C · E ′ = 1. Then L := π ∗ OX (E ′ )|D satisfies deg L = 2 and taking cohomology in 0 → π ∗ OX (E ′ − C) → π ∗ OX (E ′ ) → L → 0 we find h0 (D, L) ≥ 2. Since pa (D) ≥ 5, Riemann-Roch implies h1 (D, L) 6= 0. But then, Clifford’s inequality h0 (D, L) ≤ 2 is in fact an equality, which implies that D is hyperelliptic. In the proof of Theorem 2.1 we have seen that ϕ restricted to D induces |ωD |, which contradicts the fact that ϕ is birational. Thus, deg ϕ ≥ 2 e is a non-degenerate integral surface in P1+C 2 , we conclude and since ϕ(X) e 2C 2 ≤ deg ϕ · C 2 ≤ deg ϕ · deg ϕ(X).

On the other hand, π ∗ OX (C) is globally generated outside a finite set of points and so we find e ≤ 2C 2 deg ϕ · deg X

with equality if and only if π ∗ OX (C) is globally generated: this is clear if π is e is smooth. If π is inseparable, we consider ϕ ◦ ̟ in the e´ tale, because then X diagram (2) and obtain the same result by arguing on X (1/2) . Putting these inequalities together, we find that π ∗ OX (C) is globally generated, e = C 2 . In particular, ϕ(X) e is a surface of minimal degree deg ϕ = 2 and deg ϕ(X) and thus cut out by quadrics [E-H87]. It remains to deal with the case C 2 = 2. Then, ϕ is a possibly rational map to 3 e is a curve. A generic G ∈ |π ∗ OX (E ′ )|, P . By contradiction, assume that ϕ(X) ′ ′ where E is a half-pencil with C · E = 1, is an integral curve with pa = 1. We find deg π ∗ OX (C)|G = 2, which implies h0 (G, π ∗ OX (C)|G ) = 2 by Riemann-Roch and Clifford’s inequality. This implies that ϕ(G) is a linearly embedded P1 ⊂ P3 . e is equal to this P1 , contradicting that ϕ(X) e linearly spans P3 . Thus, But then ϕ(X) e is a surface and we conclude as before. ϕ(X)

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3. C OMPLETE

INTERSECTIONS OF THREE QUADRICS

e which turns out to In this section we study one particular birational model of X, 5 be the complete intersection of three quadrics in P . This extends results of Cossec [Co85, Section 8] and Verra [Ve83, Theorem 5.1] to characteristic 2. We explicitly e As describe the equations and the action of the finite flat group scheme G on X. a byproduct, we obtain that all Enriques surfaces in any characteristic arise via the Bombieri–Mumford–Reid construction in [B-M76, §3]. e we study linear systems |π ∗ OX (C)|, In order to find projective models of X, 2 where C is an irreducible curve with C > 0. By Theorem 2.1 and Theorem 2.4, e → PN is birational onto its image if and only the associated morphism ϕ : X 2 if Φ(C) ≥ 2 and C ≥ 4. In this case, the codimension of the image is equal to C 2 − 1. Thus, in our setup, models of smallest possible codimension are of codimension 3 in P5 . Moreover, by [C-D89, Lemma 3.6.1], irreducible curves with C 2 < 10 satisfy Φ(C) ≤ 2. We are thus led to studying irreducible curves with C 2 = 4 and Φ(C) = 2. e → P5 , Theorem 3.1. For every Enriques surface X there exists a morphism ϕ : X which is birational onto its image. More precisely, there is a Cartesian diagram ϕ

:

/ ϕ(X) e

e X

π

X

/ P5

π′

ν

/ X′

such that e is a complete intersection of three quadrics, (1) ϕ(X) (2) ν is a birational morphism and X ′ has at worst Du Val singularities, (3) π is a torsor under under a finite flat group scheme G, which arises as e → X ′ , and pull-back from a G-torsor ϕ(X) e is induced by a linear G-action of the ambient P5 . (4) the G-action on ϕ(X)

P ROOF. Let C be an irreducible curve on X with C 2 = 4 and Φ(C) = 2, which always exists by [C-D89, Chapter IV §9], but see also Proposition 3.4. We set L := OX (C). By Theorem 2.1 and Proposition 2.3, the invertible sheaf π ∗ L on e gives rise to a birational morphism, whose image ϕ(X) e is cut out by quadrics. X 0 ∗ 0 ∗ ⊗2 e e We compute h (X, π L) = 6 and h (X, π L ) = 18 via pushing forward these sheaves to X (see the proof of Theorem 2.1 details). Thus, there are three quadric e is a complete intersection of three quadrics. relations and hence ϕ(X) e induces a G-action on H 0 (X, e π ∗ L). This gives rise to Next, the G-action on X e a linear G-action on P5 extending the G-action on ϕ(X). Every irreducible curve that has zero-intersection with C is a (−2)-curve [C-D89, Proposition 4.1.1]. Since OX (C) is globally generated, big and nef, M ν : X → X ′ := Proj H 0 (X, L⊗n ) n≥0

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is a birational morphism that contracts those (−2)-curves having zero-intersection with C and nothing else. In particular, X ′ has at worst Du Val singularities. Thus, H 1 (X, OX ) ∼ = H 1 (X ′ , OX ′ ) and ωX is 2-torsion if and only if ωX ′ is. This e ′ → X ′. implies that the canonical G-torsor π arises as pull-back from a G-torsor X ′ ⊗n ⊗n Since X has only Du Val singularities, L (resp. ωX ⊗ L ) for n ≥ 0 and ν∗ (L⊗n ) (resp. ν∗ (ωX ⊗ L⊗n )) have isomorphic global sections. Thus, the graded e ′ ′∗ eπ∗ L of (X, e π ∗ L) is isomorphic to the graded ring R e′ ′∗ ring R π ν∗ L of (X , π ν∗ L). eπ∗ L , O(1)) is just ϕ(X) e ⊂ P5 by Proposition 2.3. On the other Now, (Proj R ′ hand, ν∗ L is ample on X (by the Nakai–Moisehzon criterion, see also the proof e ′ . Thus, X e ′ is of Proposition 3.4 below), and so ν∗ π ∗ L ∼ = π ′∗ ν∗ L is ample on X ′ e ∗ . isomorphic to Proj R π L e it is natural to Polarizations. Having just established a projective model of X, 5 e → P , as well as ’how far’ ϕ is from being an ask for uniqueness of ϕ : X isomorphism. Since our previous result extends work of Cossec [Co85, Section 8] and Verra [Ve83, Theorem 5.1] to characteristic 2, we define Definition 3.2. A Cossec–Verra vertible sheaf L ∈ P(X), where P(X) := L ∈ Pic(X)

polarization on an Enriques surface X is an in there exists an irreducible curve C . with C ∈ |L|, C 2 = 4, Φ(C) = 2

e → P5 corresponding to |π ∗ L| by ϕL . We denote the morphism X

Clearly, every L ∈ P(X) is big and nef. To decide whether it is ample, let us recall that an irreducible curve on X is a (−2)-curve, i.e., has self-intersection −2, if and only if it is smooth and rational. Such curves are called nodal and we denote by R(X) the set of all nodal curves. Proposition 3.3. For L ∈ P(X) the following properties are equivalent: (1) L · α > 0 for every α ∈ R(X), (2) L is ample, (3) π ∗ L is very ample, and e → ϕL (X) e is an isomorphism. (4) ϕL : X In general, the reduced exceptional locus of ϕL is the union of the reduced inverse images of all those nodal curves having zero-intersection with L. P ROOF. By [C-D89, Corollary 3.2.2], every effective divisor is linearly equivalent to one that is the positive sum of curves of arithmetic genus 1 and 0. Since Φ(C) = 2, the intersection of L with curves of arithmetic genus 1 is positive. Thus, by the Nakai–Moishezon criterion for ampleness, L is ample if and only if it has positive intersection with every nodal curve. This establishes 1 ⇔ 2. From the proof of Theorem 3.1 we get 1 ⇔ 4. The equivalence 3 ⇔ 4 is obvious. The last assertion follows from the proof of Theorem 3.1. An Enriques surface X is called unnodal if R(X) is empty. Over the complex numbers, a generic Enriques surface is unnodal [B-P83, Proposition 2.8]. Let us

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recall that a Reye congruence is the subset of G(1, 3) parametrizing lines in P3 that lie on at least two quadrics of a given generic 3-dimensional family of quadrics. By [Co83, Theorem 1], a generic nodal Enriques surfaces is a Reye congruence. This said, we establish a couple of facts about existence, uniqueness and ampleness of Cossec–Verra polarizations: Proposition 3.4. Cossec–Verra polarizations always exist. More precisely: (1) if X is unnodal, then P(X) is infinite and every L ∈ P(X) is ample, (2) if X is a generic Reye congruence and p 6= 2 then there exists an ample Cossec–Verra polarization. Moreover, if p ≥ 19 then P(X) is infinite. (3) if X is extra special then P(X) is finite and no Cossec–Verra polarization is ample. Moreover, there exist nodal curves on X, whose inverse images e are contracted by ϕL for every L ∈ P(X). on X Over the complex numbers, P(X) modulo Aut(X) is finite, and is of order 252, 960 for a generic Enriques surface. Remark 3.5. Extra special surfaces exist in characteristic 2 only and are discussed e8 -extra special surface possesses only in [C-D89, Chapter III.5]. For example, an E e is non-normal and the only ϕL partially conone Cossec–Verra polarization, X tracts the non-normal locus. We shall see in Section 5 below that this is closely related to obstructions of the deformation functor, as well as to the exceptional Enriques surfaces studied in [E-SB04]. P ROOF. Let us recall, e.g. from [C-D89, Chapter II.5] that the Enriques lattice E, i.e., the N´eron–Severi group of an Enriques surface modulo torsion, is isometric to the hyperbolic lattice corresponding to the Dynkin diagram T2,3,7 : α1

α2

α3

α4

α5

α6

α7

α8

α9

◦

◦

◦

◦

◦

◦

◦

◦

◦

◦

α0

Let WE be the Weyl group with respect to all roots and WX be the Weyl group with respect to R(X). We denote by CX := {x ∈ E | xα ≥ 0, ∀α ∈ R(X)} the nodal chamber of X. As explained in [C-D89, Chapter III.2], CX is a fundamental domain of VX := {x ∈ E | x2 ≥ 0} for the WX -action. Moreover, let VX+ be the connected component of VX containing the class of an ample divisor and set + := VX+ ∩ CX . CX ∨ Let ω1 := α∨ 1 ∈ E be the fundamental weight of the root α1 ∈ E. It follows from [C-D89, Corollary 2.5.7] that elements of P(X) correspond to those elements + . In particular, P(X) is not empty. The of the orbit Isom(E) · ω1 that lie in CX stabilizer Stab(ω1 ) is the Weyl group corresponding to the Dynkin diagram T2,3,7 with vertex α1 removed, which is of type D9 . In particular, Stab(ω1 ) is finite.

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15

+ = VX+ and so P(X) corresponds to If X is unnodal then WX is trivial, CX the cosets of WE = Isom(E)/{±id} modulo Stab(ω1 ), which is infinite. Since X is unnodal, every L ∈ P(X) is ample by Proposition 3.3. Moreover, from Stab(ω1 ) ∼ = W (D9 ), we infer W (D9 )/W (D9 )(2) ∼ = (Z/2Z)8 ⋊ S9 , see [C-D89, Proposition 2.8.4]. By [B-P83, Theorem (3.4)], a generic complex Enriques surface satisfies Aut(X) ∼ = WE (2) and thus WE /WE (2) ∼ = O + (10, F2 ) by [C-D89, Theorem 2.9.1]. This identifies P(X) modulo Aut(X) with WE /WE (2) modulo W (D9 )/W (D9 )(2), which has

220 · 35 · 52 · 7 · 17 · 31 |O+ (10, F2 )| = = 25 · 3 · 5 · 17 · 31 = 252, 960 |(Z/2Z)8 ⋊ S9 | 215 · 34 · 5 · 7 elements. Next, let X be a generic Reye congruence. It contains 10 genus-one half pencils Fi and 10 nodal curves Di such that Fi Fj = 1 and Di Dj = 2 for i 6= j, see [Co83, Lemma 3.2.1]. It follows from the proof of [Co83, Proposition 3.2.5], that the invertible sheaf corresponding to Cij := Fi + 21 (Di + Dj ) for i 6= j belongs to P(X). If X is a generic Reye congruence then the genus-one fibrations |2Fi | have no reducible fibers by the remark after [Co83, Proposition 3.2.4]. From this it is not difficult to compute that every nodal curve intersects Cij positively, i.e., the corresponding ϕL is an isomorphism. It follows from [C-D85, Theorem 1], that automorphism groups of generic nodal Enriques surfaces in characteristic p ≥ 19 + and since Stab(ω1 ) is finite, we conclude are infinite. Since this group acts on CX that P(X) is infinite. e8 -extra special, then WX = Isom(E)/h±idi, which implies that it If X is E contains only one genus-one fibration |2E| and that P(X) consists of only one element L. Then we use [C-D89, Proposition 3.6.2] to see that |L| = |2E + 2R1 + ... + 2R7 + R8 + R10 | (notation as in case 4 of [C-D89, page 185]) from which we read off that R1 , ..., R8 and R10 have zero-intersection with L. In the other extra special cases, the genus one fibrations are described in [C-D89, Chapter III.5] and applying [C-D89, Proposition 3.6.2] to a divisor class |C| with C 2 = 4, Φ(C) = 2 we end up with a finite list of possibilities of how to write |C| in terms of genus-one fibrations. First, this shows that P(X) is finite. Second, in these explicit lists we can always find nodal curves that have zero-intersection with C for any choice of C and any decomposition into genus-one pencils. We leave the lengthy, yet straight forward details to the reader. Finally, over the complex numbers, the subgroup of Isom(E) generated by WX and Aut(X) is of finite index [Do84]. In particular, there are only finitely many + ) and orbits of Isom(E) · ω1 modulo WX (needed to move the vector into CX modulo Aut(X). Thus, P(X) modulo Aut(X) is finite. Explicit equations. We end this section by determining explicit equations of the e For later use, let us extend our setup for a moment: complete intersection ϕL (X). let R be a complete, local and Noetherian ring with residue field k. Then, a finite flat group scheme of length 2 over R is isomorphic to Ga,b for some a, b ∈ R

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CHRISTIAN LIEDTKE

with ab = 2 by [O-T70, Theorem 2]. Straight forward calculations – but see also [B-M76, p.222] – show: Lemma 3.6. The regular representation of Ga,b , where ab = 2, associates to every R-algebra S the homomorphism ρreg : Ga,b (S) = {s ∈ S | s2 = as} → GL2 (S) 1 s s 7→ 0 1 − bs

If we set T := R[x1 , y1 , ..., xn , yn ] and assume that Ga,b acts on each pair xi , yi via ρreg then the following quadrics are Ga,b -invariant xi xj ,

yi2 − a xi yi ,

xi y j + y i xj + b y i y j .

Moreover, the Ga,b -invariants of T in even degree are generated by these invariant quadrics. τ ∼ Let X be an Enriques surface over k and assume that Pic (X) = Gb,a . At the beginning of Section 2 we gave an explicit description of the induced Ga,b -torsor e → X. π : X

Next, we choose a Cossec–Verra polarization L on X, and remind the reader that such polarizations always exist by [C-D89, Chapter IV §9] or Proposition 3.4. Proposition 3.7. There exists a linear Ga,b -action on P5 such that e → P5 , ϕL : X

becomes Ga,b -equivariant. Its image is the complete intersection of three quadrics. More precisely, there exist coordinates x1 , x2 , x3 , y1 , y2 , y3 on P5 such that (1) the Ga,b -action on each pair xi , yi is as in Lemma 3.6, and e are linear combinations of the (2) such that the quadrics cutting out ϕ(X) invariant quadrics of Lemma 3.6. e is a complete intersection of three P ROOF. By Theorem 3.1, the image ϕ(X) quadrics. We take cohomology in the short exact sequence 0 → L → π∗ π ∗ L → ω X ⊗ L → 0

and note that Ga,b acts via its regular representation on π∗ π ∗ L, see the discussion at the beginning of Section 2. Thus, we can choose a basis x1 , x2 , x3 of H 0 (X, L), as well as lifts y1 , y2 , y3 of a basis of H 0 (X, ωX ⊗L) to H 0 (X, π∗ π ∗ L) such that Ga,b acts on each pair xi , yi as in Lemma 3.6. In particular, as a Ga,b -representation, H 0 (X, π∗ π ∗ L) is isomorphic ρ⊕3 reg , that is, 3 copies of the regular representation. Now, consider the exact sequence of Ga,b -modules µ

0 → ker µ → Sym2 H 0 (X, π∗ π ∗ L) → H 0 (X, π∗ π ∗ (L⊗2 )) → 0 .

The kernel ker µ is easily seen to be 3-dimensional. Arguing as above, we see that H 0 (X, π∗ π ∗ L⊗2 ) is isomorphic to ρ⊕9 reg as Ga,b -representation. Decomposing the Ga,b -representation on Sym2 H 0 (X, π∗ π ∗ L), we find that Ga,b acts trivially on e is cut out by three quadrics, all of which are Ga,b -invariant. ker µ. Thus, ϕ(X)

ENRIQUES SURFACES

17

Remark 3.8. Following an idea of Reid, Bombieri and Mumford [B-M76, §3] gave the first construction of all three types of Enriques surfaces in characteristic 2. Our result shows that in fact all Enriques surfaces arise in this way - after possibly resolving Du Val singularities of the quotient. 4. M ODULI

AND

L IFTING –

THE

P OLARIZED C ASE

In this section we consider Enriques surfaces together with ample Cossec–Verra polarizations. Given an Enriques surface, such a polarization always exist after possibly contracting nodal curves to Du Val singularities. Thus, we study pairs (X, L), where X is an Enriques surface with at worst Du Val singularities and L is an ample Cossec-Verra polarization. We show that such pairs have an extremely nice deformation theory, construct their moduli space MCV,ample and prove lifting to characteristic zero. Let us first slightly extend Definition 3.2: an invertible sheaf L on an Enriques surface X ′ with at worst Du Val singularities is a Cossec–Verra polarization if ν ∗ L on X is, where ν : X → X ′ denotes the minimal desingularization. In particular, if L is a Cossec–Verra polarization on a (smooth) Enriques surface then M ν : X → X ′ := Proj H 0 (X, L⊗n ) n≥0

is a contraction and OX ′ (1) is an ample Cossec–Verra polarization on X ′ . In this section, k denotes an algebraically closed field of characteristic p ≥ 0 and R is a complete, local and Noetherian ring with residue field k. Picard scheme and effectivity. Before studying deformations, moduli and lifting, we have to understand extensions of invertible sheaves on formal deformations. Proposition 4.1. Let X be an Enriques surface with at worst Du Val singularities over k and X → Spf R be a formal deformation of X. If L is an invertible sheaf on X and (1) if X is classical, then L extends uniquely to X , and (2) if X is non-classical, then L⊗2 extends to X . Moreover, if L1 and L2 are ⊗2 ⊗2 extensions of L to R then L1 ∼ = L2 . In particular, every formal deformation is effective. P ROOF. If X is classical then h1 (OX ) = h2 (OX ) = 0 and so the first assertion is a standard result in deformation theory. If X is non-classical then char(k) = 2. We write X as limit Xn → Spec Rn with X = X0 and where each Rn+1 → Rn is a small extension. Thus, for every n ≥ 0 we have an exponential sequence × × → 0, → OX 0 → OX → OX n n+1

and, taking successive extensions, we end up with short exact sequences (3)

× × → 0. → OX 0 → Kn,s → OX n n+s

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CHRISTIAN LIEDTKE

We claim that Kn,s , considered as sheaf of Abelian groups, is a direct sum of Witt vectors of finite length over OX : first of all, we have Kn,1 ∼ = OX for all n. Moreover, interpreting Xn+s as X(n+1)+(s−1) , the respective sequences (3) yield a short exact sequence 0 → Kn+1,s−1 → Kn,s → OX → 0 .

By induction on s we may assume that Kn+1,s−1 is a direct sum of sheaves of Witt vectors of finite length. Considering each of its summand individually, Lemma 4.3 reveals that also Kn,s is of this form. This establishes our claim. × ). Taking cohomology Now, L corresponds to an element in Pic(X) ∼ = H 1 (OX 2 in (3), we denote by δs the coboundary map Pic(X) → H (K0,s ). Then δs (L) 6= 0 if and only if L extends to Xs . Since K0,s is a direct sum of Wm OX ’s and since H 2 (Wm OX ) ∼ = k for every m by Lemma 4.3, we conclude that H 2 (K0,s ) is 2-torsion. Thus, for every s, the obstruction to extending L⊗2 to Xs vanishes, proving that L⊗2 extends to X . Moreover, if L1 and L2 are extensions of L to Xs , then their “difference” lies in H 1 (K0,s ), which is 2-torsion by the same reasoning as for H 2 (K0,s ) above. This ⊗2 ⊗2 implies L1 ∼ = L2 . Finally, if L is ample on X, then so is L⊗2 and since the latter extends to X , the formal deformation is effective by Grothendieck’s existence theorem. Remark 4.2. The classical case is trivial. In case X is singular and R = W (k), extension of L⊗4 has been shown in [La83, Theorem 1.4]. Lemma 4.3. Let X be a non-classical Enriques surface over k. Then Ext1 (OX , Wm OX ) ∼ and H 2 (Wm OX ) ∼ = H 1 (Wm OX ) ∼ = k = k.

Moreover, the only non-trivial extension class corresponds to Wm+1 OX . P ROOF. We have H 1 (W OX ) = 0 and H 2 (W OX ) ∼ = k by [Il79, Section II.7.3]. Thus, the m-fold Verschiebung V m yields an exact sequence Vm

0 → H 1 (Wm OX ) → H 2 (W OX ) → H 2 (W OX ) → H 2 (Wm OX ) → 0. Since H 1 (Wm OX ) ∼ = Ext1 (OX , Wm OX ) is contained in k and contains the nontrivial class [Wm+1 OX ], we conclude H 1 (Wm OX ) ∼ = k for all m. In particular, this class generates the cohomology group. Counting dimensions in the above exact sequence, we infer H 2 (Wm OX ) ∼ = k for all m.

Proposition 4.4. Under the previous assumptions, Picτ (X /R) is a finite flat group scheme of length 2 over R.

P ROOF. By [B-M76, Theorem 2], Picτ of an Enriques surface over any field is finite of rank 2, and so Picτ (X /R) is quasi-finite. Choosing an ample invertible sheaf L on X and extending L⊗2 to X , which is possible by our previous result, we see that X → Spec R is projective. Thus, R is Noetherian, and X → Spec R is smooth, projective and with geometrically irreducible fibers, which implies that Picτ (X /R) is projective [FGA, no 236, Corollaire 4.2]. Thus, Picτ (X /R) is in fact a finite group scheme over R. As Picτ of the special fiber has rank 2, Nakayama’s

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19

lemma implies that the R-module H 0 (Spec R, Picτ (X /R)) is generated by 2 elements. If ωX 6∼ = OX then these two invertible sheaves define two distinct morphisms from Spec R to Picτ (X /R). Thus, the closure of the union of their images defines a non-trivial finite flat subgroup scheme. By the previous discussion this group scheme has to coincide with Picτ (X /R). We may thus assume ωX ∼ = OX . Let us write X as limit over Xn → Rn , where the Rn ’s are local Artin algebras. Clearly, we have f∗ OXn = Rn and using Grothendieck duality, we find R2 f∗ OXn ∼ = Rn . By flatness, we = R 2 f ∗ ω Xn ∼ P i i have i (−1) length(R f∗ OXn ) = χ(OX ) = 1. Thus, the length of R1 f∗ OXn as Rn -module is equal to the length of Rn as Rn -module. From H 1 (OX ) ∼ = k and 1 Nakayama’s lemma we find that R f∗ OXn is a cyclic Rn -module. This implies that R1 f∗ OXn ∼ = Rn . Now, R1 f∗ OXn is the tangent space to Pic(Xn /Rn ) at the zero-section. Since this is locally free of rank 1, we infer the existence of a finite flat and infinitesimal group scheme of rank 2 inside Pic0 (Xn /Rn ). Since this holds for arbitrary n, we conclude that the limit of these group schemes coincides with Picτ (X /R). Let us recall once more that every finite flat group scheme of rank 2 over R is of the form Ga,b for some a, b ∈ R with ab = 2, see [O-T70, Theorem 2], as well as Lemma 3.6. The following result extends Proposition 3.7 to families: Proposition 4.5. Let L be a Cossec–Verra polarization on X and assume that it extends to X → Spec R. If Picτ (X /R) ∼ = Gb,a , then there exists a linear Ga,b action on P5R , as well as a Ga,b -equivariant morphism ϕ : Xe → P5R ,

whose image is the complete intersection of three quadrics. Moreover, there exist coordinates x1 , x2 , x3 , y1 , y2 , y3 on P5R such that (1) the Ga,b -action is as in Lemma 3.6, and (2) such that the quadrics cutting out ϕ(Xe) are R-linear combinations of the invariant quadrics of Lemma 3.6.

P ROOF. We just established that Picτ (X /R) is a finite and flat group scheme of rank 2 over R, say, isomorphic to Gb,a . By [Ra70, Proposition (6.2.1)], there exists a finite flat Ga,b -torsor π : Xe → X . From the proof of Proposition 4.4 we infer a short exact sequence 0 → OX → π∗ OXe → ωX → 0 .

Now, let L be an extension of L to X . Since h1 (X, L) = 0 by [C-D89, Theorem 1.5.1], global sections of L extend to global sections of L. Clearly, ωX /R ⊗ L extends ωX ⊗ L and h1 (X, ωX ⊗ L) shows that its global sections extend, too. Thus, f∗ L and f∗ (ωX ⊗ L) are free R-modules of rank 3. In particular, π ∗ L defines a map ϕL : Xe 99K P5R that coincides with ϕL on the special fiber. By Proposition 3.7, ϕL is a morphism whose image is the complete intersection of three quadrics and so the same is true for ϕL by openness

20

CHRISTIAN LIEDTKE

of these properties. As in the proof of Proposition 3.7 we conclude that ϕL is Ga,b -equivariant. Moreover, since the three quadrics in the special fiber cutting out e are Ga,b -invariant, the same is true for the quadrics cutting out ϕ (Xe). ϕL (X) L Deformations and lifting. Now, we come to one of our main results. Let X be an Enriques surface with at worst Du Val singularities over k and L be an invertible sheaf on X. We define the functor local Artin algebras Def X,L : → ( Sets ) with residue field k

that associates to each R the set of pairs (X , L), where X is a flat deformation of X over R and L extends L to X . By Proposition 4.4, Picτ (X /R) is a finite and flat group scheme of length 2 over R. We denote by Def Grp,2 the functor that assigns to each local Artin algebra R with residue field k the set of finite flat group schemes of rank 2 over R. Such group schemes are of the form Ga,b for some a, b ∈ R with ab = 2. Thus, Def Grp,2 has W (k)[[a, b]]/(ab − 2) as pro-representable hull. Theorem 4.6. Let X be an Enriques surface with at worst Du Val singularities together with an ample Cossec–Verra polarization L. Then the morphism of functors Def X,L → Def Grp,2 ,

that assigns to each flat deformation X /R its Picτ (X /R), is smooth.

Remark 4.7. We shall see in Section 5 that this fails to be true if L is not ample or if we consider unpolarized deformations. P ROOF. Existence of this morphism follows from Proposition 4.4. Now, let R′ → R be a small extension, let (X , L) be a deformation of (X, L) over R and let G ′ be a finite flat group scheme extending G := Picτ (X /R) to R′ . To prove smoothness, we have to find an extension of (X , L) to R′ whose relative Picτ is isomorphic to G ′ . By Proposition 4.5, there exists a G D -torsor π : Xe → X and π ∗ (L) defines an embedding into P5R , whose image is a complete intersection of three G D -invariant quadrics. From the explicit description in Lemma 3.6, we see that we can find a G ′D -action on P5R′ , as well as a complete intersection of G ′D -invariant quadrics Xe′ , extending Xe together with its G D -action on P5R to R′ . Next, consider the morphism Ψ : P5R′ → P11 R′

given by the G ′D -invariant quadrics of Lemma 3.6. By the same lemma, the G ′D invariants of even degree in R′ [x1 , y1 , ..., x3 , y3 ] are generated by these 12 invariant quadrics. Thus, Ψ is the quotient morphism by G ′D . In particular, X ′ := Ψ(Xe′ ) ∼ = Xe′ /G ′D is flat over Spec R′ , extending Xe/G D ∼ = X to R′ . Finally, the G ′D -action defines a descent data on OP5 ′ (1)|Xe′ . Thus, by finite flat R descent, it comes from an invertible sheaf L′ on X ′ , which extends L. As a direct consequence, we obtain lifting to characteristic zero - we note that the ramification needed for the lifting is entirely controlled by Picτ .

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21

Theorem 4.8. Let X be an Enriques surface with at worst Du Val singularities. Assume that X admits an ample Cossec–Verra polarization. (1) If X is not supersingular then it lifts over W (k), √ and (2) if X is supersingular then it lifts over W (k)[ 2], but not over W (k). ∼ Ga,b . If X is not P ROOF. We fix a, b ∈ k with ab = 2 such that Picτ (X) = ′ ′ supersingular then a 6= 0 or b 6= 0 and we can find a , b ∈ W (k) with a′ b′ = 2 mapping to a, b, and thus a lift of Ga,b to W (k). Lifting of X over W (k) then follows by running through the proof of Theorem 4.6 with R = k and R′ = W (k). Of course, R′ → R is not a small extension but the proof also works in this case. Alternatively, Theorem 4.6 provides us with a formal lifting of X over Spf W (k), which is algebraizable by Proposition 4.1. √ If X√is supersingular, then a = b = 0 and Gπ,π with π := 2 lifts G0,0√∼ = α2 to W (k)[ 2]. The same arguments as before show that X lifts over W (k)[ 2]. On the other hand, if X were a lifting of X over W (k) then Picτ (X /R) would be a finite flat group scheme of length 2 over W (k) with special fiber α2 by Proposition 4.4. However, this contradicts the fact that αp does not admit liftings over W (k), see also [O-T70, Theorem 2]. Moduli spaces. For an algebraically closed field k of characteristic p ≥ 0, we consider the set MCV,ample of pairs (X, L), where - X is an Enriques surfaces with at worst Du Val singularities over k, and - L is an ample Cossec–Verra polarization on X. We end this section by giving this set the structure of an Artin stack and describing its geometry: Theorem 4.9. MCV,ample is a quasi-separated Artin stack of finite type over k. (1) If p 6= 2 then MCV,ample is irreducible, unirational, 10-dimensional and smooth over k. (2) If p = 2 then MCV,ample consists of two irreducible, unirational, smooth and 10-dimensional components µ2 MCV,ample

and

Z/2Z

MCV,ample .

Moreover, - they intersect transversally along an irreducible, unirational, smooth α2 and 9-dimensional closed substack MCV,ample , α2 - MCV,ample parametrizes supersingular surfaces, α2 G - MCV,ample − MCV,ample parametrizes singular surfaces (G = µ2 ) and classical surfaces (G = Z/2Z), respectively, G contains an open and dense substack, whose - for all G, MCV,ample geometric points correspond to smooth surfaces. P ROOF. We only discuss the case p = 2, since the analysis for p 6= 2 is analogous to the case of singular Enriques surfaces in characteristic 2. First of all, since for every (X, L) ∈ MCV,ample , X is projective and since every formal deformation is effective by Proposition 4.1, the set MCV,ample can

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be given the structure of a quasi-separated Artin stack of finite type over k, see [Ar74a, Example (5.5)] for a sketch, or [Ri96] for a detailed discussion. Next, let G be a finite flat group scheme of length 2 and consider the GD -action on P5 as in Lemma 3.6. Let Ψ : P5 → P11 be the morphism defined by the GD invariant quadrics as in the proof of Theorem 4.6. As explained in [B-M76, §3], the inverse image of a generic hyperplane section yields the canonical double cover of an Enriques surface together with a GD -action. Such surfaces are overparametrized by an open dense subset UG of the Grassmannian Grass(3, 12). It follows from Proposition 3.7, that all Enriques surfaces with at worst Du Val singularities, ample Cossec–Verra polarization and whose canonical double cover is a GD -torsor arise this way. If we denote by N G the substack of surfaces with Picτ ∼ = G, then we G have just shown that UG maps dominantly onto N , showing irreducibility, as well as unirationality. Now, if X → S is a family of Enriques surfaces with at worst Du Val singularities, then the set of points such that Xs is a classical Enriques surface is open, since the property h1 (OX ) = 0 is open by semi-continuity. Also, the set of points s.th. Xs is singular is open: by [Ar74b] there exists a surjective map S ′ → S and an algebraic space X ′ → S ′ that simultaneously resolves the singularities of X → S. But then, the property of being a singular Enriques surface, i.e., satisfying h0 (ΩX ) = 0, is open on S ′ by semi-continuity. Now, for each s ∈ S, the h → Oh map on Henselizations OS,s S ′ ,s′ is finite. We conclude that being a singular Enriques surface is stable under generization also on S, proving openness. Similar arguments show that the set of points such that Xs is supersingular, is closed. We conclude that N G for G = µ2 and for G = Z/2Z belong to different components G of MCV,ample . We denote these components by MCV,ample and remark that they G contain N as open and dense substacks. Next, let (X, L) ∈ N α2 . The group scheme G0,t over k[[t]] has special fiber α2 and generic fiber µ2 . By Theorem 4.6, there exists a family X → Spec k[[t]] with special fiber X and Picτ (X /k[[t]]) ∼ = G0,t . In particular, the geometric generic fiber of this family is a singular Enriques surface. This shows that N α2 is a closed µ2 substack of MCV,ample . Using Gt,0 instead, we conclude that N α2 is also a closed

µ2 and MCV,ample intersect along N α2 . substack of MCV,ample . Thus, MCV,ample The functor Def Grp,2 restricted to the subcategory of local Artin k-algebras has pro-representable hull k[[x, y]]/(xy), see the discussion before Theorem 4.6. But then, the statements about smoothness and transversal intersections follow from Theorem 4.6. Generic hyperplane sections of Ψ(P5 ) ⊆ P11 (notation as in Theorem 4.6) yield smooth singular, classical and supersingular Enriques surfaces with ample Cossec–Verra polarizations, see also [B-M76, §3]. Since ampleness and smoothG for ness are open properties, there exist open and dense substacks of MCV,ample G = µ2 , Z/2Z, and α2 , respectively, corresponding to smooth surfaces. We partly postpone the computation of the dimension to Section 5: namely, for G = Z/2Z and µ2 we shall prove there that the just-established open and G parametrizing smooth surfaces are isomorphic to dense substacks of MCV,ample Z/2Z

Z/2Z

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23

open substacks of the yet to be defined stack MCV,smooth . Thus, it will suffice to compute the dimension at such points of MCV,smooth , which is equal to 10, see the discussion after Proposition 5.3. From this, the dimension and the local description α2 of MCV,ample follows from Theorem 4.6 and the description of Def Grp,2 . Remark 4.10. Over the complex numbers, Casnati [Ca04] considered Enriques surfaces together with invertible sheaves with self-intersection 4, and showed that the corresponding moduli space is rational. Clearly, not every such polarization is Cossec–Verra. Nevertheless, in view of this result it would be interesting to know whether the components of MCV,ample are rational. Clearly, we do not have to restrict ourselves to deformations over k-algebras. We leave it to the reader to use the previous proof to show that in fact Theorem 4.11. The moduli space MCV,ample is a quasi-separated Artin stack of finite type and smooth over Spec Z[x, y]/(xy − 2). 5. U NPOLARIZED

MODULI

In this section we study deformations, moduli and liftings of smooth Enriques surfaces - with and without Cossec–Verra polarizations, leading to the moduli spaces MCV,smooth and MEnriques . The general picture is similar to the one of the previous section. These two moduli spaces are related to MCV,ample via Artin’s functor that simultaneously resolves Du Val singularities in families. At points corresponding to classical and exceptional Enriques surfaces, it is this functor which is responsible for the non-smoothness of MEnriques and MCV,smooth . Let us consider the following sets: (1) MEnriques , whose elements are smooth Enriques surfaces. (2) MCV,ample , whose elements are pairs (X, L), where - X is an Enriques surface with at worst Du Val singularities, and - L is an ample Cossec–Verra polarization. (3) MCV,smooth , whose elements are pairs (X, L), where - X is a smooth Enriques surface, and - L is a Cossec–Verra polarization. (4) MGrp,2 , whose elements are finite flat group schemes of length 2. Clearly, all these sets carry structures of algebraic stacks, but let us defer the discussion of quasi-separatedness until Theorem 5.2 below. In any case, they are related by two functors: MCV,smooth m Φcont mmmm m m mm v mm m

MCV,ample

The first functor contracts nodal curves: Φcont : MCV,smooth → (X, L)

QQQ QQΦ QQforget QQQ QQ(

MEnriques

MCV,ample L 7 → (X ′ := Proj n≥0 H 0 (X, L⊗n ), OX ′ (1))

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Now, a morphism from some base scheme S to MCV,ample corresponds to a family X → S of Enriques surfaces with at worst Du Val singularities together with an ample Cossec–Verra polarization L. Thus, its fiber is simultaneous resolutions of singularities of −1 ′ Φcont (S) : S /S 7→ X ×S S ′ → S ′ together with the pullback of L Artin [Ar74b] showed that this functor is representable by a locally quasi-separated and quasi-finite algebraic space over S. Thus, Proposition 5.1. The functor Φcont is representable, locally quasi-separated and a bijection on geometric points. By Theorem 4.9, every component of MCV,ample contains an open and dense substack over which Φcont is in fact an isomorphism. As an application we obtain quasi-separatedness of our stacks. We note that this is in contrast to K3 surfaces, where the moduli space of unpolarized surfaces is highly non-separated. Theorem 5.2. The moduli spaces MEnriques , MCV,ample and MCV,smooth are quasi-separated Artin stacks of finite type over k. P ROOF. We have to verify that the assumptions of [Ar74a, Theorem (5.3)] are fulfilled. As explained in [Ar74a, Example (5.5)], this is clear for MCV,ample , see also Theorem 4.9. In the remaining cases, everything except quasi-separatedness is clear. But now, quasi-separatedness of MCV,ample together with Proposition 5.1 implies quasi-separatedness of MCV,smooth . Finally, since MCV,smooth → Spec k factors over MEnriques → Speck, we obtain quasi-separatedness of MEnriques . The second functor forgets the Cossec–Verra polarization Φforget : MCV,smooth → MEnriques (X, L)

7→

X

Here, we have the following: Proposition 5.3. The functor Φforget is representable, separated and locally finite and flat. It is smooth at a geometric point (X, L) if and only if X is classical. P ROOF. A family X → S of Enriques surfaces is automatically projective by Proposition 4.1. By [FGA, no 232, Theorem 3.1], Pic(X /S) is representable by a separated scheme, which is locally of finite type over S. The condition L2 = 4 is closed and such invertible sheaves lie discrete in the N´eron–Severi lattice. From Theorem 2.1 and Theorem 2.4 we see that the Φ ≥ 2 is characterized by the property that ϕL is birational. Since this is an open property, so is the condition Φ ≥ 2. On the other hand, invertible sheaves with selfintersection number 4 satisfy Φ ≤ 2 by [C-D89, Lemma 3.6.1], and thus Φ = 2 is an open property for such invertible sheaves. We conclude that Φ−1 forget (S) is τ locally represented by Pic (X /S), proving local finiteness and flatness of Φforget . In particular, Φforget is smooth at (X, L) if and only if Picτ (X) is reduced, which is the case if and only if X is classical.

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25

An invertible sheaf L on a smooth surface X determines via d log a class in H 1 (X, ΩX ), the Chern class of L, which can also be interpreted as an extension class in Ext1 (ΘX , OX ), the Atiyah extension of L 0 → OX → AL → ΘX → 0 .

The groups H i (X, ΘX ) provide a tangent-obstruction theory for MEnriques , and the groups H i (X, AL ) a tangent-obstruction theory for MCV,smooth . Then, the differential of Φforget can be computed from the cohomology sequence dΦforget

... → H 1 (OX ) → H 1 (AL ) −→ H 1 (ΘX ) → H 2 (OX ) → ... If X is classical, then dΦforget is an isomorphism, whereas it has one-dimensional kernel and cokernel in the non-classical case. In particular, we find h0 (AL ) = h0 (ΘX ) + 1,

h1 (AL ) = h1 (ΘX ),

and

h2 (AL ) = h2 (ΘX ).

The hi (ΘX ) are well-known, see Section 1. In particular, if (X, L) is is neither supersingular nor exceptional, MCV,smooth is 10-dimensional at (X, L). Using the previously defined functors, we now compare our moduli spaces: Theorem 5.4. In characteristic 6= 2, all three stacks MEnriques , MCV,smooth and MCV,ample are smooth, irreducible, unirational and 10-dimensional. P ROOF. We have shown this for MCV,ample in Theorem 4.9. There exists an open and dense substack, over which MCV,smooth and MCV,ample are isomorphic, showing irreducibility and unirationality of MCV,smooth . The latter implies that MEnriques is irreducible and unirational. Smoothness of MCV,smooth and MEnriques follows from h2 (ΘX ) = h2 (AL ) = 0. Remark 5.5. Using analytic methods, Kond¯o [Ko94] has shown that MEnriques over the complex numbers is rational. It would be interesting to know whether this is also true in positive characteristic. We described the geometry of MCV,ample in characteristic 2 in Theorem 4.9, and extend this now to the other moduli stacks. Theorem 5.6. Let M be one of MEnriques , MCV,ample and MCV,smooth . Then it consists of two 10-dimensional, irreducible and unirational components M µ2

and

M Z/2Z .

Moreover, - they intersect along a closed substack M α2 , which is 9-dimensional, irreducible and unirational, - M α2 parametrizes supersingular Enriques surfaces, - M G − M α2 parametrizes singular (G = µ2 ) and classical (G = Z/2Z) Enriques surfaces, respectively. The local geometry is a little bit more tricky and given by the following result

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Theorem 5.7. Let X be an Enriques surface in characteristic 2 and L a Cossec– Verra polarization. - If X is singular, or classical and not exceptional, then all three moduli stacks are smooth at X, (X, L), and Φcont (X, L), respectively. - If X is classical and exceptional, then - MCV,ample is smooth at Φcont (X, L), whereas - MCV,smooth and MEnriques are not smooth at (X, L) and X. More precisely, they acquire locally irreducible hypersurface singularities. - If X is supersingular, then the intersection of M µ2 and M Z/2Z in X is transversal - for Φcont (X, L) ∈ MCV,ample , - for (X, L) ∈ MCV,smooth if L is ample, and - for X ∈ MEnriques , if X admits an ample Cossec–Verra polarization. The latter two conditions hold along an open and dense substack of M α2 . Remarks 5.8. Let us note that (1) we do not know whether the intersection at supersingular points is always transversal, and that (2) although generic Enriques surfaces satisfy h0 (ΘX ) = 0, exceptional and supersingular Enriques surfaces fulfill h0 (ΘX ) = 1. This implies that MEnriques cannot be a Deligne–Mumford stack in characteristic 2. P ROOF (of both theorems). The assertions on dimension, irreducibility and unirationality are shown as in the proof of Theorem 5.4. We established all stated properties of MCV,ample in Theorem 4.9. Moreover, smoothness in the singular or classical and non-exceptional case follows as in the proof of Theorem 5.4. If X is a classical and exceptional Enriques surface, then h1 (AL ) − h0 (AL ) = h1 (ΘX ) − h0 (ΘX ) = 11. However, the moduli spaces are 10-dimensional at this point, and so MEnriques and MCV,smooth cannot be smooth at X and (X, L), respectively. Since the obstruction spaces H 2 (ΘX ) and H 2 (AL ) are 1-dimensional, the singularities are hypersurface singularities. As Φcont is a bijection on geometric points, and MCV,ample is smooth, we conclude local irreducibility. Now, let X be supersingular. Transversal intersection of two components at Φcont (X, L) ∈ MCV,ample has been established in Theorem 4.9. Using Φcont and Φforget , we conclude that also the other two moduli spaces consist of two components intersecting along the supersingular locus. Now, if L is ample then MCV,smooth and MCV,ample are locally isomorphic near (X, L), and thus the intersection is transversal. Let us now prove transversally at X ∈ MEnriques in case X admits an ample Cossec–Verra polarization L: then (X, L) is a geometric point of MCV,ample . By Theorem 4.6, MCV,ample → MGrp,2 is smooth and in particular, every first-order deformation of α2 can be extended to a first-order deformation of the pair (X, L). Forgetting L, we infer that the differential of the map MEnriques → MGrp,2 at X is surjective. Given a deformation X → S of S, and a small extension S → S ′ such that Picτ (X /S) cannot be extended to S ′ , also X → S cannot be extended to S ′ by Proposition 4.4. Thus, the map of obstruction spaces ob(MEnriques ) → ob(MGrp,2 ) is non-trivial. Both spaces

ENRIQUES SURFACES

27

are 1-dimensional, and so the map is in particular injective. This implies that MEnriques → MGrp,2 is in fact smooth at X. Again, we conclude that the intersection of the components at X is transversal. Remark 5.9. At a point of MEnriques corresponding to a singular and exceptional Enriques surface X, the following happens: There exists a family X → S over some local base with special fiber X, as well as a small extension S → S such that the family cannot be extended over S. After choosing a Cossec–Verra polarization L on X, this polarization extends uniquely to X (since X is classical), and Φcont yields a family X ′ → S. Since MCV,ample ′ is smooth at Φcont (X, L), the family X ′ → S extends to a family X → S. By construction, X → S is a simultaneous resolution of singularities of X ′ → S over ′ S. By assumption, a simultaneous resolution of singularities of X → S extending X → S does not exist over S. However, it does exist after a ramified extension of S by Artin’s result [Ar74b]. Summing up, the singularities of MEnriques at X can be explained via ADEcurves and obstructions coming from Artin’s simultaneous resolution functor. Over the complex numbers, similar phenomena have been described in [B-W74]. Let us finally address the lifting problem for Enriques surfaces. Every Enriques surface X admits a Cossec–Verra polarization L. After contracting those nodal curves that have zero-intersection with L, we obtain an Enriques surface X ′ with at worst Du Val singularities. In Theorem 4.8 we have shown √ that X ′ lifts over W (k) unless it is supersingular, in which case it lifts over W (k)[ 2]. Here is what we know about lifting of smooth Enriques surfaces: Theorem 5.10. Let X be an Enriques surface in positive characteristic p. (1) If p 6= 2 then X lifts over W (k). (2) If X is a singular, or a classical and non-exceptional Enriques surface, then it lifts over W (k). (3) If X is supersingular √ and admits an ample Cossec–Verra polarization then it lifts over W (k)[ 2], but not over W (k). (4) In the remaining cases, X lifts over a possibly ramified extension of W (k). P ROOF. In the first two cases, we have h2 (ΘX ) = 0, i.e., there exists a formal lifting over √ W (k), whose algebraization follows from Proposition 4.1. Lifting over W (k)[ 2] and non-lifting over W (k) for supersingular surfaces admitting ample Cossec–Verra polarizations has been established in Theorem 4.8. In the remaining cases, Theorem 5.6 tells us that the deformation functor has only locally irreducible hypersurface singularities, i.e., there exists a formal lifting over a ramified extension of W (k), which is algebraizable by Proposition 4.1. Remark 5.11. The first two results are well-known, see [La83] and [E-SB04]. It would be interesting to know whether exceptional classical √ surfaces lift over W (k) and, whether all supersingular surfaces lift over W (k)[ 2].

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D EPARTMENT OF M ATHEMATICS , S TANFORD U NIVERSITY, 450 S ERRA M ALL , S TANFORD CA 94305-2125, USA E-mail address: [email protected]