An outline of the log minimal model program for the moduli space of curves

TL;DR

This paper explores the log minimal model program for the moduli space of stable curves, predicting critical values where the model changes and analyzing the stability of certain curves to outline the process.

Contribution

It provides a conjectural formula for critical values in the log minimal model program using intersection theory and GIT stability analysis.

Findings

Derived a formula for critical alpha values in the log MMP

Analyzed GIT stability of curves with tails and bridges

Outlined the sequence of model changes in the log MMP

Abstract

Hassett and Keel predicted that there is a descending sequence of critical $\alpha$ values where the log canonical model for the moduli space of stable curves with respect to $\alpha \delta$ changes. We derive a conjectural formula for the critical values in two different ways, by working out the intersection theory of the moduli space of hyperelliptic curves and by computing the GIT stability of certain curves with tails and bridges. The results give a rough outline of how the log minimal model program would proceed, telling us when the log canonical model changes and which curves are to be discarded and acquired at the critical steps.