Hyper-Kähler fourfolds and Kummer surfaces

Abstract

We show that a Hilbert scheme of conics on a Fano fourfold double cover of PP2xP2 ramified along a divisor of bidegree (2,2) admits a ℙ 1‐fibration with base being a hyper‐Kähler fourfold. We investigate the geometry of such fourfolds relating them with degenerated EPW cubes (see Iliev et al., J. reine angew. Math. (2016), https://doi.org/10.1515/crelle‐2016‐0044), with elements in the Brauer groups of 𝐾3‐surfaces of degree 2, and with Verra threefolds. These hyper‐Kähler fourfolds admit natural involutions and complete the classification of geometric realizations of antisymplectic involutions on hyper‐Kähler 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..