On two families of Enriques categories over K3 surfaecs

Abstract: 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, 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 involutions on an O'Grady's hyperk\"ahler tenfold.