Moduli of Enriques surfaces Klaus Hulek March 4, 015 This is joint work with V. Gritsenko. We work mostly over C but there will be a few exceptions at the end. 1 Introduction Let S be an Enriques surface over C, i.e., ω S O S but ω S = O S and q(s) = 0. Let X S denote the étale double cover by a K3 surface. We have ρ(s) = 1 e(x) = 1 and b (S) = 10. The Néron-Severi group of S is NS(S) U E 8 ( 1) Z/Z where U is a hyperbolic plane and E 8 is associated with the corresponding root system. The Z/Z is generated by the class of ω S. Consider a marking of S Recall with involution admitting eigenspaces ϕ : H (S, Z)/torsion U E 8 ( 1). H (X, Z) 3U E 8 ( 1) =: L K3 ϱ : L K3 L K3 (x, y, z, u, v) (z, y, x, v, u) Eig(ϱ) + = {(x, 0, x, u, u), x U, u E 8 ( 1)} = U() E 8 ( ) =: M and Eig(ϱ) = {(x, y, x, u, u)} U U() E 8 ( ) =: N. 1
The signature of N is (, 10). Fact: Given ϕ we can find a marking ϕ : H (X, Z) L K3 such that ϱ ϕ = ϕ σ where σ is the involution on X. Note that ϕ(p H (S, Z)) Eig(ϱ) + = M. Since σ acts on H,0 (X) by 1 we find Then we can define a period domain ϕ(ω X ) N = Eig(ϱ). Ω N = {[x] P(N C) : (x, x) = 0, (x, x) > 0} = D N D N and we may assume ϕ(ω X ) D N. Let O(N) denote the integral orthogonal group and O + (N) those isometries preserving D N. We take M En = O + (N)\D N, which is quasi-projective of dimension ten. For each root δ N (with δ = ) we have a hyperplane H δ := {[x] : (x, δ) = 0} M En. Theorem 1 (Horikawa, Namikawa) Associating to an Enriques surface a period point as above gives a bijection between M 0 En := M En \ and Enriques surfaces up to isomorphism. Remark 1 This is not a moduli space in the sense of representing a functor of Enriques surfaces. Liedtke has studied technical issues with defining this as a stack. A polarization or other additional data is needed. Theorem (Kondo) M En is rational.
Moduli spaces Definition 1 A (semi-)polarized Enriques surface is a pair (S, L) where L is a (semi-)ample line bundle. A numerically polarized Enriques surface is a pair (S, L) where L is ample and L Num(S). Viehweg has contructed quasi-projective coarse moduli spaces M a En,h of polarized Enriques surfaces with a given type of polarization. Definition The type of a polarization is an O(M) = O(M( 1 ))-orbit of a primitive vector h U E 8 ( 1) = M( 1). Notation. We use M to denote moduli spaces, and M to denote orthogonal modular varieties. We have an involution ι : M a En,h M a En,h (S, L) (S, L ω S ) and set M num En,h := M a En,h/ ι. 3 Groups and orthogonal modular varieties Given a lattice L, let D(L) denote the discriminant group with its standard quadratic form. Note that D(M) = D(N) = F 10 and O(D(M)) = O(D(N)) = O + (F 10 ) is the orthogonal group of even type which has order 1 3 5 5 7 17 31. Fix h U E 8 ( 1) = M( 1 ) primtive and Set O(M, h) = {g O(M), g(h) = h}. Õ(N) Γ h = π 1 N (π M(O(M, h)) O(N) so that Γ + h = Γ h O + (N) acts on D N. Here π M : O(M) O(D(M)) and similarly for π N. 3
Let M En,h = Γ + h \D N M En,h where the typersurface,h is the inverse image of. It is irreducible. An element v N, v = 4 belongs to one of two O + (N) orbits. It is even or odd respectively depending on whether (v, N) is Z or Z. Even ( 4)-vectors are of the form (δ, δ) for δ U E 8 ( 1) when thinking of the embedding of U E 8 ( 1) into Eig(ρ). The ( 4)-vectors define a hypersurface 4 M En, the nodal Enriques surfaces, with preimage Let 4,h M En,h. 4,h 4,h consist of those irreducible components where δ h = 0. We have spaces M num En,h = M En,h \ (,h 4,h ). Theorem 3 There is a degree étale double cover which identifies M num En,h with Mnum En,h. M a Eh,h M num En,h Remark 1. The open part MEn,h num which is given by removing all of corrresponds to non-nodal polarized Enriques surfaces. 4,h. The points on 4,h \ can be interpreted as semi-polarized Enriques surfaces. Question 1 1. Is M a En,h always connected?. Is M a En,h an orthogonal modular variety? Corollary 1 There are only finitely many isomorphism classes of moduli spaces of (primitively) polarized Enriques surfaces. 4
Ingredients of the proof: 1. There are only finitely many groups Õ(N) Γ h O(N).. 4,h has finitely many components. 3. MEn,h num has only finitely many étale double covers since this is a finite CW complex and thus H 1 (MEn,h num, Z/Z) is finite. Corollary The Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative for h < 3. Corollary 3 There are moduli space of numerically polarized Enriques surface birational to the moduli space with -level structure M En () = Õ+ (N)\D N which is of general type (Gritsenko, in preparation). 4 Arithmetic questions Question Over which fields can we construct these moduli spaces? For the Cossec-Verra presentation, Liedtke showed the moduli space is defined over Q. Also, Kondo has constructed a projective model of M En () P 185 given by 3 3 5 7 17 31 quartic equations. This embedding is given via automorphic forms and the relation can be understood on terms of representation theory. Question 3 Over which field is M En () defined? This is a projective variety of general type which might be of arithmetic interest. 5