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K-moduli of Fano threefolds and genus four curves

Yuchen Liu, Junyan Zhao

TL;DR

The article determines the K-moduli space of Fano threefolds in deformation family №2.15, identifying it as a two-step birational modification of the GIT moduli of $(3,3)$-curves on ${\mathbb P}^1\times{\mathbb P}^1$ and proving its realization as a Hassett–Keel model for $\overline{M}_4$. The authors implement the moduli continuity method using lattice-polarized K3 surface geometry, “general elephants,” and Sarkisov links to classify all K-(semi/poly)stable members, including degenerations to singular cubic threefolds. Their results connect explicit K-stability classifications with the birational geometry of genus four curves, providing a concrete moduli interpretation and a robust framework for analyzing similar families of Fano varieties.

Abstract

In this article, we study the K-moduli space of Fano threefolds obtained by blowing up $\mathbb{P}^3$ along $(2,3)$-complete intersection curves. This K-moduli space is a two-step birational modification of the GIT moduli space of $(3,3)$-curves on $\mathbb{P}^1 \times \mathbb{P}^1$. As an application, we show that our K-moduli space appears as one model of the Hassett--Keel program for $\overline{M}_4$. In particular, we classify all K-(semi/poly)stable members in this deformation family of Fano varieties. We follow the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients.

K-moduli of Fano threefolds and genus four curves

TL;DR

The article determines the K-moduli space of Fano threefolds in deformation family №2.15, identifying it as a two-step birational modification of the GIT moduli of -curves on and proving its realization as a Hassett–Keel model for . The authors implement the moduli continuity method using lattice-polarized K3 surface geometry, “general elephants,” and Sarkisov links to classify all K-(semi/poly)stable members, including degenerations to singular cubic threefolds. Their results connect explicit K-stability classifications with the birational geometry of genus four curves, providing a concrete moduli interpretation and a robust framework for analyzing similar families of Fano varieties.

Abstract

In this article, we study the K-moduli space of Fano threefolds obtained by blowing up along -complete intersection curves. This K-moduli space is a two-step birational modification of the GIT moduli space of -curves on . As an application, we show that our K-moduli space appears as one model of the Hassett--Keel program for . In particular, we classify all K-(semi/poly)stable members in this deformation family of Fano varieties. We follow the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients.
Paper Structure (19 sections, 49 theorems, 114 equations)

This paper contains 19 sections, 49 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. K3 surfaces
  4. Geometry of Fano threefolds
  5. K-stability and K-moduli spaces
  6. Variation of GIT for $(2,3)$-complete intersection curves
  7. Moduli of curves and Hassett--Keel program
  8. Moduli spaces of K3 surfaces
  9. Geometry and moduli of K3 surfaces
  10. K3 surfaces in anticanonical linear series
  11. K-semistable limits of a one-parameter family
  12. Volume comparison and general elephants
  13. Modifications on the family and degree 6 K3 surfaces
  14. Birational model as a singular cubic threefold
  15. K-moduli, VGIT, and the Hassett–Keel program
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mathscr{M}^K_{\textup{№2.15}}$ be the K-moduli stack of the family №2.15, and $\overline{M}^K_{\textup{№2.15}}$ be its good moduli space. Then the followings hold.

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2: cf. May72SD
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • ...and 93 more