Hyperkähler fourfolds and Kummer surfaces

Abstract

We show that a Hilbert scheme of conics on a Fano fourfold double cover of ${\mathbb{P}}^2\times {\mathbb{P}}^2$ ramified along a divisor of bidegree $(2,2)$ admits a ${\mathbb{P}}^1$-fibration with base being a hyper-Kähler fourfold. We investigate the geometry of such fourfolds relating them with degenerated EPW cubes, with elements in the Brauer groups of $K3$ surfaces of degree $2$, and with Verra threefolds studied in [Ver04]. These hyper-Kähler fourfolds admit natural involutions and complete the classification of geometric realizations of anti-symplectic involutions on hyper-Kähler $4$-folds of type $K{3}^{[2]}$. As a consequence we present also three constructions of quartic Kummer surfaces in ${\mathbb{P}}^3$: as Lagrangian and symmetric degeneracy loci and as the base of a fibration of conics in certain threefold quadric bundles over ${\mathbb{P}}^1$.

to appear in Proceedings of the London Mathematical Society

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• Degenerations of Gushel-Mukai fourfolds, with a view towards irrationality proofs
• Projective models of Nikulin orbifolds
• Derived-natural automorphisms on Hilbert schemes of points on generic K3 surfaces