The final log canonical model of $\bar{M}_6$

TL;DR

This paper describes a specific birational model of the moduli space of genus 6 curves, showing it as the last significant step in its log minimal model program and providing a new upper bound for its moving slope.

Contribution

It identifies the final log canonical model of ar{M}_6 as quadric hyperplane sections of a degree 5 del Pezzo surface, extending previous work on genus 4.

Findings

Identifies the last non-trivial space in the log minimal model program for ar{M}_6

Provides a new upper bound for the moving slope of the moduli space

Describes the birational model via quadric hyperplane sections of a del Pezzo surface

Abstract

We describe the birational model of $\bar{M}_6$ given by quadric hyperplane sections of the degree 5 del Pezzo surface. In the spirit of the genus 4 case treated by Fedorchuk, we show that it is the last non-trivial space in the log minimal model program for $\bar{M}_6$. We also obtain a new upper bound for the moving slope of the moduli space.