Wall crossing for K-moduli spaces of plane curves

TL;DR

This paper develops a framework for understanding how moduli spaces of certain Fano pairs change as a parameter varies, with applications to plane curves and connections to GIT and K3 surface moduli.

Contribution

It constructs proper K-moduli spaces for log Fano pairs and establishes a wall-crossing framework, linking these spaces to GIT, Hacking's compactification, and K3 moduli.

Findings

K-moduli spaces are isomorphic to GIT moduli for small coefficients

First wall crossing results in weighted blow-ups of Kirwan type

Complete wall crossing descriptions for degrees 4, 5, 6

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.