Moduli of Genus Six Curves and K-stability

TL;DR

This paper explores the K-moduli spaces of genus six curves embedded in quintic del Pezzo surfaces, analyzing their structure, wall-crossing phenomena, and relationships with GIT and K3 surface moduli.

Contribution

It provides a detailed classification of genus six curves in K-moduli and describes the wall-crossing structure, connecting different moduli space constructions.

Findings

Classified strata of genus six curves in K-moduli.

Described wall-crossing behavior of K-moduli spaces.

Connected K-moduli with GIT and K3 surface moduli.

Abstract

The K-moduli theory provides a different compactification of moduli spaces of curves. As a general genus six curve can be canonically embedded into the smooth quintic del Pezzo surface, we study in this paper the K-moduli spaces $\overline{M}^K(c)$ of the quintic log Fano pairs. We classify the strata of genus six curves $C$ appearing in the K-moduli by explicitly describing the wall-crossing structure. The K-moduli spaces interpolate between two birational moduli spaces constructed by GIT and moduli of K3 surfaces via Hodge theory.