Hyperk\"ahler fourfolds and Kummer surfaces

TL;DR

This paper explores the geometry of hyperk"ahler fourfolds derived from Fano fourfolds, relating them to Kummer surfaces, EPW cubes, and Verra threefolds, and classifies involutions on these fourfolds.

Contribution

It introduces new hyperk"ahler fourfolds from Fano fourfolds, links them with known geometric objects, and classifies anti-symplectic involutions on hyper-K"ahler fourfolds of type K3^{[2]}.

Findings

Hyperk"ahler fourfolds admit a fibration with a hyper-K"ahler fourfold base.

Connections established with degenerated EPW cubes, Brauer groups of K3 surfaces, and Verra threefolds.

Constructed quartic Kummer surfaces via Lagrangian degeneracy loci, symmetric degeneracy loci, and conic fibrations.

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-K\"{a}hler 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-K\"{a}hler fourfolds admit natural involutions and complete the classification of geometric realizations of anti-symplectic involutions on hyper-K\"{a}hler $4$-folds of type $K3^{[2]}$. 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 $\mathbb{P}^1$.