Abstract. We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the eleven dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on MorrisonNikulin involutions, these are Nikulin involutions which interchange two copies of E 8 (−1) in the Néron Severi group.In his paper [Ni1] Nikulin started the study of finite groups of automorphisms on K3 surfaces, in particular those leaving the holomorphic two form invariant, these are called symplectic. He proves that when the group G is cyclic and acts symplectically, then G ∼ = Z/nZ, 1 ≤ n ≤ 8. Symplectic automorphisms of K3 surfaces of orders three, five and seven are investigated in the paper [GS]. Here we consider the case of G ∼ = Z/2Z, generated by a symplectic involution ι. Such involutions are called Nikulin involutions (cf.[Mo, Definition 5.1]). A Nikulin involution on the K3 surface X has eight fixed points, hence the quotientȲ = X/ι has eight nodes, by blowing them up one obtains a K3 surface Y . In the paper [Mo] Morrison studies such involutions on algebraic K3 surfaces with Picard number ρ ≥ 17 and in particular on those surfaces whose Néron Severi group contains two copies of E 8 (−1). These K3 surfaces always admit a Nikulin involution which interchanges the two copies of E 8 (−1). We call such involutions Morrison-Nikulin involutions. The paper of Morrison motivated us to investigate Nikulin involutions in general. After a study of the maps on the cohomology induced by the quotient map, in the second section we show that an algebraic K3 surface with a Nikulin involution has ρ ≥ 9 and that the Néron Severi group contains a primitive sublattice isomorphic with E 8 (−2). Moreover if ρ = 9 (the minimal possible) then the following two propositions are the central results in the paper: Proposition 2.2. Let X be a K3 surface with a Nikulin involution ι and assume that the Néron Severi group NS(X) of X has rank nine. Let L be a generator of E 8 (−2) ⊥ ⊂ NS(X) with L 2 = 2d > 0 and let Λ 2d := ZL ⊕ E 8 (−2) (⊂ NS(X)).Then we may assume that L is ample and:(1) in case L 2 ≡ 2 mod 4 we have Λ 2d = NS(X);
We study a proposal of D'Hoker and Phong for the chiral superstring measure for genus three. A minor modification of the constraints they impose on certain Siegel modular forms leads to a unique solution. We reduce the problem of finding these modular forms, which depend on an even spin structure, to finding a modular form of weight 8 on a certain subgroup of the modular group. An explicit formula for this form, as a polynomial in the even theta constants, is given. We checked that our result is consistent with the vanishing of the cosmological constant. We also verified a conjecture of D'Hoker and Phong on modular forms in genus 3 and 4 using results of Igusa.
Abstract. We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.
Abstract. A main issue in superstring theory are the superstring measures. D'Hoker and Phong showed that for genus two these reduce to measures on the moduli space of curves which are determined by modular forms of weight eight and the bosonic measure. They also suggested a generalisation to higher genus. We showed that their approach works, with a minor modification, in genus three and we announced a positive result also in genus four. Here we give the modular form in genus four explicitly. Recently S. Grushevsky published this result as part of a more general approach.
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