An Outline of the Log Minimal Model Program for the Moduli Space of Curves

Abstract

ABSTRACTWe consider the log minimal model program for the moduli space of stable curves. It is widely believed now that there is a descending sequence of critical $α$ values where the log canonical model for the moduli space of stable curves with respect to $αδ$ changes, where $δ$ denotes the divisor of singular curves. 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, predicting when the log canonical model changes and which curves are to be discarded and acquired at the critical steps.