Hyper-Kahler fourfolds and Kummer surfaces

Abstract

We show that a Hilbert scheme of conics on a Fano fourfold double cover of P2xP2 ramified along a divisor of bidegree (2,2) admits a P1-fibration with base being a hyper-Kahler fourfold. We investigate the geometry of such fourfolds relating them with degenerated EPW cubes (see Iliev et al., J. reine angew. Math. (2016), ), with elements in the Brauer groups of K3-surfaces of degree 2, and with Verra threefolds. These hyper-Kahler fourfolds admit natural involutions and complete the classification of geometric realizations of antisymplectic involutions on hyper-Kahler fourfolds of type K3[2]. As a consequence we present also three constructions of quartic Kummer surfaces in P3: as Lagrangian and symmetric degeneracy loci and as the base of a fibration of conics in certain threefold quadric bundles over P1.