On two families of Enriques categories over K3 surfaecs

This article studies the moduli spaces of semistable objects related to two families of Enriques categories over K3 surfaces, coming from quartic double solids and special Gushel-Mukai threefolds. In particular, some classic geometric constructions are recovered in a modular way, such as the double EPW sextic and cube associated with a general Gushel-Mukai surface, and the Beauville’s birational involution on the Hilbert scheme of two points on a quartic K3 surface. In addition, we describe the singular loci in some moduli spaces of semistable objects and an explicit birational involution on O’Grady’s hyperkähler tenfold. Also, the appendix investigates the general theory of Enriques categories over K3 surfaces and provides a criterion for when an equivariant category of a K3 surface is an Enriques category.

💡 Research Summary

The paper investigates two families of Enriques categories that arise from quartic double solids and special Gushel‑Mukai threefolds, focusing on the geometry of moduli spaces of semistable objects. An Enriques category is defined as a triangulated category equivalent to the equivariant derived category (D^b(S)^\Pi) of a K3 surface (S) under a non‑symplectic involution (\Pi). The author restricts to “geometric” Enriques categories, i.e. admissible subcategories of derived categories of smooth projective varieties that inherit a (\mathbb Z/2)-action.

A central technical tool is Theorem 1.1 (originally from