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On two families of Enriques categories over K3 surfaecs

Ziqi Liu

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, 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.

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, 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.

Paper Structure

This paper contains 47 sections, 94 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Enriques categories over K3 surfaces
  4. Kuznetsov components of quartic double solids
  5. Kuznetsov components of special Gushel--Mukai threefolds
  6. More on Enriques categories over K3 surfaces
  7. Conventions and organization
  8. Acknowledgments
  9. Geometric Enriques categories over K3 surfaces
  10. Group actions and equivariant objects
  11. Reconstruction theorem
  12. Geometric Enriques categories over K3 surfaces
  13. Stability conditions on geometric triangulated categories
  14. Stability conditions
  15. The complex manifold of stability conditions
  16. ...and 32 more sections

Key Result

Theorem 1.1

PPZ23 Consider a geometric Enriques category $\mathcal{E}$ over a K3 surface $S$ characterized by the involutions ${\textup{U}}$ on $\mathcal{E}$ and $\Pi$ on ${\textbf{D}}^b(S)$, a ${\textup{U}}$-fixed proper stability condition $\sigma_{\mathcal{E}}$ on $\mathcal{E}$, and the induced proper stabil where the target is the fixed locus of the ${\textup{U}}$-induced action on $\mathfrak{M}_{\sigma_{

Theorems & Definitions (178)

  • Theorem 1.1
  • Theorem 1.2: Subsection \ref{['QDS-ss-over-u1']}
  • Theorem 1.3: Subsection \ref{['QDS-ss-over-2mu1']}
  • Theorem 1.4: Subsection \ref{['QDS-ss-over-mu2']}
  • Theorem 1.5: Subsection \ref{['QDS-ss-over-2u1']}
  • Theorem 1.6: Subsection \ref{['QDS-ss-u1minusu2']}
  • Theorem 1.7: Subsection \ref{['GM-ss-over-w1']}
  • Theorem 1.8: Subsection \ref{['GM-ss-over-w2']}
  • Remark 1.9
  • Theorem 1.10: Subsection \ref{['GM-ss-over-w1plusw2']}
  • ...and 168 more