Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Wall crossing for K-moduli spaces of plane curves

Kenneth Ascher, Kristin DeVleming, Yuchen Liu

Abstract

We construct proper good moduli spaces parametrizing K-polystable $\mathbb{Q}$-Gorenstein smoothable log Fano pairs $(X, cD)$, where $X$ is a Fano variety and $D$ is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as $c$ varies. The main application in this paper is the case of plane curves of degree $d \geq 4$ as boundary divisors of $\mathbb{P}^2$. In this case, we show that when the coefficient $c$ is small, the K-moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K-moduli spaces are weighted blow-ups of Kirwan type. We also describe all wall crossings for degree 4,5,6, and relate the final K-moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

Wall crossing for K-moduli spaces of plane curves

Abstract

We construct proper good moduli spaces parametrizing K-polystable -Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K-moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K-moduli spaces are weighted blow-ups of Kirwan type. We also describe all wall crossings for degree 4,5,6, and relate the final K-moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

Paper Structure

This paper contains 52 sections, 114 theorems, 254 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. K-stability of log Fano pairs
  4. Valuative criteria for K-stability
  5. Normalized volumes
  6. CM line bundles
  7. Good moduli spaces in the sense of Alper
  8. Hacking's compact moduli of plane curves
  9. Construction of K-moduli spaces of smoothable log Fano pairs
  10. Foundations
  11. Continuity method
  12. K-semistable thresholds are constructible
  13. Properness
  14. Almost log Calabi-Yau cases
  15. Stabilization of quotient stacks
  16. ...and 37 more sections

Key Result

Theorem \oldthetheorem

Let $\chi_0$ be the Hilbert polynomial of an anti-canonically polarized Fano manifold. Fix $r\in\mathbb{Q}_{>0}$ and a rational number $c\in (0,\min\{1,r^{-1}\})$. Then there exists a reduced Artin stack $\mathcal{K}\mathcal{M}_{\chi_0,r,c}$ of finite type over $\mathbb{C}$ parametrizing all K-semis

Figures (1)

  • Figure 1: Log Fano wall crossings for K-moduli spaces of plane quintics

Theorems & Definitions (248)

  • Theorem \oldthetheorem: =Theorem \ref{['thm:lwxlog']}
  • Theorem \oldthetheorem: =Theorem \ref{['thm:logFano-wallcrossing']}
  • Theorem \oldthetheorem: First wall crossing
  • Theorem \oldthetheorem: =Theorem \ref{['thm:highdim']}
  • Theorem \oldthetheorem: $d = 4, 6$, see Theorem \ref{['thm:quartsext']} and Section \ref{['sec:K3surface']}
  • Theorem \oldthetheorem: $d = 5$, see Theorems \ref{['thm:firstwallps']}, \ref{['thm:secondwall']} and Section \ref{['sec:quintics']}
  • Theorem \oldthetheorem: =Theorem \ref{['thm:projectivity']}
  • Conjecture \oldthetheorem: Log Calabi-Yau wall crossings, see Conjecture \ref{['conj:logCY']}
  • Remark \oldthetheorem: Postscript
  • Definition \oldthetheorem
  • ...and 238 more