The geometry of the ball quotient model of the moduli space of genus four curves

TL;DR

This paper links Kondo's ball quotient compactification of genus four curves with a GIT quotient of the Chow variety, providing explicit descriptions and resolving the birational map through a blow-up, thus offering a modular interpretation of boundary points.

Contribution

It explicitly describes the GIT quotient related to Kondo's space and resolves the birational map via a blow-up, connecting geometric invariant theory with the moduli space of genus four curves.

Findings

The GIT quotient coincides with Kondo's compactification.

The birational map is resolved by a single blow-up.

Provides a modular interpretation of boundary points.

Abstract

S. Kondo has constructed a ball quotient compactification for the moduli space of non-hyperelliptic genus four curves. In this paper, we show that this space essentially coincides with a GIT quotient of the Chow variety of canonically embedded genus four curves. More specifically, we give an explicit description of this GIT quotient, and show that the birational map from this space to Kondo's space is resolved by the blow-up of a single point. This provides a modular interpretation of the points in the boundary of Kondo's space. Connections with the slope nine space in the Hassett-Keel program are also discussed.