Double EPW sextics associated to Gushel-Mukai surfaces

TL;DR

The paper establishes a precise smoothness criterion for double EPW sextics arising from 2-dimensional Gushel-Mukai surfaces by linking geometric properties of Brill-Noether general $\langle 10\rangle$-polarized $K3$ surfaces to the period and Lagrangian data of EPW sextics. It uses the Debarre–Kuznetsov correspondence to translate GM data into EPW data and derives divisorial conditions in moduli spaces that govern smoothness, showing the double cover $X_A$ is a hyper-Kähler manifold when the associated $K3$ surface contains neither lines nor quintic elliptic pencils. The work then constructs symplectic automorphism actions on families of smooth double EPW sextics and provides sharp lattice-theoretic bounds on automorphism groups of GM varieties, including higher-dimensional cases, thereby elucidating how automorphisms lift through the EPW/GMs framework. Overall, the results advance understanding of the interplay between GM geometry, EPW sextics, and hyper-Kähler structures, with concrete implications for automorphism groups and moduli.

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-Kähler manifolds. We also provide bounds for the automorphism group of Gushel-Mukai varieties in dimension 2 and higher.

Theorems & Definitions (63)

• Theorem 1.1
• Definition 1.2
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Corollary 2.6
• Theorem 2.7