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K-stability of Thaddeus' moduli of stable bundle pairs on genus two curves

Junyan Zhao

Abstract

The moduli space of bundle stable pairs $\overline{M}_C(2,Λ)$ on a smooth projective curve $C$, introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of $\overline{M}_C(2,Λ)$ to that of the moduli spaces of stable vector bundles $\overline{N}_C(2,Λ)$. In particular, Thaddeus' moduli spaces for genus two curves are all K-stable.

K-stability of Thaddeus' moduli of stable bundle pairs on genus two curves

Abstract

The moduli space of bundle stable pairs on a smooth projective curve , introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of to that of the moduli spaces of stable vector bundles . In particular, Thaddeus' moduli spaces for genus two curves are all K-stable.

Paper Structure

This paper contains 27 sections, 55 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. K3 surfaces
  4. Geometry of Fano threefolds
  5. K-stability and K-moduli spaces
  6. Moduli space of bundle stable pairs
  7. Moduli spaces of K3 surfaces
  8. Geometry and moduli of K3 surfaces
  9. K3 surfaces in anticanonical linear series
  10. K-semistable limits of a one-parameter family
  11. Volume comparison and general elephants
  12. Modifications on the family and degree 8 K3 surfaces
  13. Birational model as a complete intersection of two quadrics
  14. Variation of GIT for lines in quartic del Pezzo threefolds
  15. VGIT of lines in quartic del Pezzo threefolds
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\mathscr{M}^K_{\textup{№2.19}}$ be the K-moduli stack of the family №2.19, and $\overline{M}^K_{\textup{№2.19}}$ be its good moduli space. Then the followings hold.

Theorems & Definitions (112)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2: cf. May72SD
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 102 more