Abstract

The parameter space for lines on a smooth cubic hypersurface in projective 5-space is a hyperkaehler 4-fold deformation equivalent to the Hilbert square of a K3 surface. By varying the cubic we get almost all isomomorphism classes of deformations of the Hilbert square of a K3 surface equipped with a polarization of Beauville-Bogomolov square 6 and divisbility 2 (these are the two discrete invariants of a primitive polarization on a deformation of the Hilbert square of a K3). There is an analogous picture if we consider deformations of the Hilbert square of a K3 surface equipped with a polarization of Beauville-Bogomolov square 2 (divisibility is necessarily equal to 1). There exists a family of sextic hypersurface in projective 5-space (EPW-sextics) with 2-dimensional singular set, which come equipped with a double cover ramified only over the singular set, such that the family of such double covers parametrizes (up to isomomorphism) almost all deformations of the Hilbert square of a K3 surface equipped with a polarization of Beauville-Bogomolov square 2. We will discuss the geometry of double EPW -sextics, in particular the period map.