Dual Double EPW-sextics and Their Periods

Abstract

Eisenbud, Popescu and Walter have constructed certain singular sextic hypersurfaces (EPW-sextics) in P (Example (9.3) of [4]) which come provided with a natural double cover: we have shown [13] that the generic such double cover is a deformation of the Hilbert square of a K3 and that the family of double EPWsextics is a locally versal family of projective deformations of (K3). Thus the family of double EPW-sextics is similar to the family of Fano varieties of lines on a cubic 4-fold (see [2]), with the following difference: the Plucker ample divisor on the Fano variety of lines has square 6 for the Beauville-Bogomolov quadratic form (see [1, 2]) while the natural polarization of a double EPWsextic has square 2 (see [13]). Let Y ⊂ P be a generic EPW-sextic: we proved in [13] that the dual Y ∨ ⊂ (P5)∨ is another generic EPW-sextic. Thus we may associate to the natural double cover X of Y a “dual”variety X∨ namely the natural double cover of Y ∨. This construction defines a (rational) involution on the moduli space of double EPW-sextics. In [13] we showed that a generic EPW-sextic is not self-dual and hence the involution on the moduli space of double EPW-sextics is not the identity; in this paper we determine the relation between the periods of a double EPW-sextic and its dual. Before stating the result we recall the definition of EPW-sextics. Let V be a 6-dimensional C-vector space. Choose an isomorphism vol : ∧ V ∼ −→ C and let ω be the symplectic form on ∧V defined by wedge product followed by vol. Let P(V ) be the projective space of 1-dimensional sub vector spaces of V ; then ω gives ∧V ⊗OP(V ) the structure of a symplectic vector-bundle of rank 20. Let F be the sub-vector-bundle of ∧V ⊗OP(V ) whose fiber F[v] over [v] ∈ P(V ) consists of tensors divisible by v: