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Hassett--Keel Program in genus four

Kenneth Ascher, Kristin DeVleming, Yuchen Liu, Xiaowei Wang

TL;DR

This work completes the Hassett–Keel program for genus four by constructing and gluing a sequence of α-stable moduli stacks \\overline{\\mathcal{M}}_4(\\alpha) with projective good moduli spaces \\overline{M}_4(\\alpha) across nine critical walls, culminating in a full modular interpretation of the log M-models for g=4. The authors deploy a triad of tools: wall-crossing for boundary-polarized Calabi–Yau pairs (bpCY) to model replacements of singular loci, KSBA stability for canonical-semistable degenerations, and K-stability to connect these with the KSBA and Chow-quotient pictures. A key advance is the stackified Chow quotient construction that realizes the hyperelliptic flip in genus four via a global moduli framework, together with a BP CY–driven local model around the C_{2A5}, C_D, and Γ loci. They also establish the smoothness of the bpCY moduli stack and identify its good moduli space with Kondō’s ball quotient compactification, linking algebraic and Hodge-theoretic moduli perspectives. Combining these ingredients yields a complete, projective, modular understanding of the genus-four log M-models and their wall-crossings, with a path toward generalizing the hyperelliptic flip to higher genus.

Abstract

The Hassett--Keel program seeks to give a modular interpretation to the steps of the log minimal model program of $\overline{\mathcal{M}}_g$. The goal of this paper is to complete the Hassett--Keel program in genus four, supplementing earlier results of Casalaina-Martin--Jensen--Laza and Alper--Fedorchuk--Smyth--van der Wyck. The main tools we use are wall crossing for moduli spaces of pairs in the sense of K-stability and KSBA stability, and the recently constructed moduli spaces of boundary polarized Calabi--Yau surface pairs. We also give a construction of the hyperelliptic flip using a stackified Chow quotient which is expected to generalize to higher genus.

Hassett--Keel Program in genus four

TL;DR

This work completes the Hassett–Keel program for genus four by constructing and gluing a sequence of α-stable moduli stacks \\overline{\\mathcal{M}}_4(\\alpha) with projective good moduli spaces \\overline{M}_4(\\alpha) across nine critical walls, culminating in a full modular interpretation of the log M-models for g=4. The authors deploy a triad of tools: wall-crossing for boundary-polarized Calabi–Yau pairs (bpCY) to model replacements of singular loci, KSBA stability for canonical-semistable degenerations, and K-stability to connect these with the KSBA and Chow-quotient pictures. A key advance is the stackified Chow quotient construction that realizes the hyperelliptic flip in genus four via a global moduli framework, together with a BP CY–driven local model around the C_{2A5}, C_D, and Γ loci. They also establish the smoothness of the bpCY moduli stack and identify its good moduli space with Kondō’s ball quotient compactification, linking algebraic and Hodge-theoretic moduli perspectives. Combining these ingredients yields a complete, projective, modular understanding of the genus-four log M-models and their wall-crossings, with a path toward generalizing the hyperelliptic flip to higher genus.

Abstract

The Hassett--Keel program seeks to give a modular interpretation to the steps of the log minimal model program of . The goal of this paper is to complete the Hassett--Keel program in genus four, supplementing earlier results of Casalaina-Martin--Jensen--Laza and Alper--Fedorchuk--Smyth--van der Wyck. The main tools we use are wall crossing for moduli spaces of pairs in the sense of K-stability and KSBA stability, and the recently constructed moduli spaces of boundary polarized Calabi--Yau surface pairs. We also give a construction of the hyperelliptic flip using a stackified Chow quotient which is expected to generalize to higher genus.
Paper Structure (37 sections, 91 theorems, 138 equations, 6 figures, 1 table)

This paper contains 37 sections, 91 theorems, 138 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method of proof
  3. Wall crossing for pairs
  4. The hyperelliptic flip and the stackified Chow quotient
  5. Preliminaries
  6. Stacks of curves
  7. Good moduli spaces for algebraic stacks
  8. Stacks from first three steps of Hassett--Keel program
  9. Chow stability and VGIT
  10. Chow stability
  11. Chow stability and VGIT in genus $4$
  12. Moduli of pairs and log Calabi--Yau wall crossing
  13. Curves on quadric cones
  14. Hyperelliptic curves and ribbons
  15. Hyperelliptic curves
  16. ...and 22 more sections

Key Result

Theorem 1.1

For each rational number $\alpha \in [\frac{5}{9},\frac{2}{3})$, there exists a smooth algebraic stack $\overline{\mathcal{M}}_4(\alpha)$ of finite type with affine diagonal that parameterizes $\alpha$-stable curves of genus four, which are projective curves of genus four with lci singularities and

Figures (6)

  • Figure 1: A depiction of the curves $C_{2A_5}$ and $C_{D}$.
  • Figure 2: A depiction of the curves $C_{A,B}$.
  • Figure 3: A depiction of the wall crossing $\overline{M}_4(\frac{5}{9} -\epsilon) \rightarrow \overline{M}_4(\frac{5}{9}) \leftarrow \overline{M}_4(\frac{5}{9}+\epsilon)$ near the curve $\Gamma$.
  • Figure 4: The construction of $S_{A_5}$.
  • Figure 5: The surface $S_{2A_5}$.
  • ...and 1 more figures

Theorems & Definitions (202)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3: Previous expectations
  • Theorem 1.4: see Theorem \ref{['thm:bpCY-is-smooth']}
  • Theorem 1.5: see Theorem \ref{['thm:sX_g-smooth']}
  • Theorem 1.6: see Theorem \ref{['thm:sX_g-gms']}
  • Theorem 1.7: see Theorem \ref{['thm:sX-gms']}
  • Definition 2.1: alper
  • Definition 2.2
  • Theorem 2.3: AHR20
  • ...and 192 more