On the double EPW sextic associated to a Gushel-Mukai fourfold

TL;DR

This paper explores when a double EPW sextic associated with a Gushel-Mukai fourfold is birational to a moduli space of sheaves on a K3 surface, linking geometric properties to solutions of a Pell equation.

Contribution

It establishes a criterion connecting the birationality of the double EPW sextic to the Hodge-speciality of the Gushel-Mukai fourfold via Pell equation solvability.

Findings

Double EPW sextic is birational to a K3 surface's Hilbert scheme under specific conditions.

Birationality depends on the Hodge-speciality and Pell equation solutions.

Provides a criterion for when the associated double cover relates to moduli spaces.

Abstract

In analogy to the case of cubic fourfolds, we discuss the conditions under which the double cover $\tilde{Y}_A$ of the EPW sextic hypersurface associated to a Gushel-Mukai fourfold is birationally equivalent to a moduli space of (twisted) stable sheaves on a K3 surface. In particular, we prove that $\tilde{Y}_A$ is birational to the Hilbert scheme of two points on a K3 surface if and only if the Gushel-Mukai fourfold is Hodge-special with discriminant $d$ such that the negative Pell equation $\mathcal{P}_{d/2}(-1)$ is solvable in $\mathbb{Z}$.