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The boundary of K-moduli of prime Fano threefolds of genus twelve

Anne-Sophie Kaloghiros, Yuchen Liu, Andrea Petracci, Junyan Zhao

Abstract

We study the K-moduli stack of prime Fano threefolds of genus twelve, known as $V_{22}$. We prove that its boundary, which parametrizes singular members, is purely divisorial and consists of four irreducible components corresponding to the four families of Prokhorov's one-nodal $V_{22}$. A key ingredient is a modular relation between Fano threefolds $X$ and their anticanonical K3 surfaces $S$. We prove that the forgetful morphism from the moduli of Fano--K3 pairs $(X,S)$ where $X$ is a K-semistable degeneration of $V_{22}$ to the moduli space of genus $12$ polarized K3 surfaces $(S,{-K_X}|_S)$ is an open immersion. In particular, the K-moduli of $V_{22}$ is governed by the moduli of their anticanonical K3 surfaces, providing a modular realization of Mukai's philosophy. Along the way, we develop a general deformation framework for Fano threefolds of large volume, which may be useful beyond the study of K-moduli.

The boundary of K-moduli of prime Fano threefolds of genus twelve

Abstract

We study the K-moduli stack of prime Fano threefolds of genus twelve, known as . We prove that its boundary, which parametrizes singular members, is purely divisorial and consists of four irreducible components corresponding to the four families of Prokhorov's one-nodal . A key ingredient is a modular relation between Fano threefolds and their anticanonical K3 surfaces . We prove that the forgetful morphism from the moduli of Fano--K3 pairs where is a K-semistable degeneration of to the moduli space of genus polarized K3 surfaces is an open immersion. In particular, the K-moduli of is governed by the moduli of their anticanonical K3 surfaces, providing a modular realization of Mukai's philosophy. Along the way, we develop a general deformation framework for Fano threefolds of large volume, which may be useful beyond the study of K-moduli.

Paper Structure

This paper contains 35 sections, 83 theorems, 219 equations, 1 figure.

Table of Contents

  1. Introduction
  2. K-moduli of $V_{22}$ and Mukai's philosophy
  3. General deformation framework for K-moduli of Fano threefolds
  4. Outline of the proof
  5. History and prior work
  6. History of $V_{22}$
  7. K-moduli of Fano threefolds
  8. Conventions and notations
  9. Preliminaries
  10. Geometry of Fano threefolds
  11. Moduli of K3 surfaces
  12. K-stability and K-moduli theory
  13. Deformations of Fano threefolds with large volume
  14. Very ampleness of anti-canonical divisors
  15. Beauville type results
  16. ...and 20 more sections

Key Result

Theorem 1.1

Every singular K-semistable $V_{22}$ is a degeneration of one-nodal $V_{22}$. In particular, the boundary of $\mathcal{M}^{\mathrm{K}}$ parametrizing singular members is purely divisorial and consists of four irreducible components, each corresponding to one of the four families of one-nodal $V_{22}

Figures (1)

  • Figure 1: The triangular prism (left) and its dual bipyramid (right).

Theorems & Definitions (175)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4: RS09
  • Definition 2.5
  • Theorem 2.6: One-nodal $V_{22}$; Pro16
  • Remark 2.7
  • ...and 165 more