Double EPW sextics associated to Gushel-Mukai surfaces

TL;DR

This paper characterizes when the double EPW sextic associated with a Gushel-Mukai surface is smooth, enabling the construction of symplectic actions on certain hyper-Kähler manifolds and providing bounds on automorphism groups.

Contribution

It offers a characterization of smooth double EPW sextics linked to Gushel-Mukai surfaces and explores symplectic actions and automorphism bounds for these geometric structures.

Findings

Characterization of smooth double EPW sextics associated to Gushel-Mukai surfaces

Construction of symplectic actions on hyper-Kähler manifolds

Bounds for automorphism groups of Gushel-Mukai varieties

Abstract

Works by O'Grady allow to associate to a 2-dimensional Gushel-Mukai variety, which is a K3 surface, a double EPW sextic. We characterize the K3 surfaces whose associated double EPW sextic is smooth. As a consequence, we are able to produce symplectic actions on some families of smooth double EPW sextics which are hyper-K\"ahler manifolds. We also provide bounds for the automorphism group of Gushel-Mukai varieties in dimension 2 and higher.