 
Summary: SINGULARITIES WITH GmACTION AND
THE LOG MINIMAL MODEL PROGRAM FOR Mg
JAROD ALPER, MAKSYM FEDORCHUK, AND DAVID SMYTH
ABSTRACT. We give a precise formulation of the modularity principle for the log canoni
cal models Mg() := Proj d0 (M g,d(KM g
+ )). Assuming the modularity principle
holds, we develop and compare two methods for determining the critical values at which a
singularity or complete curve with Gmaction arises in the modular interpretation of Mg().
The first method involves a new invariant of curve singularities with Gmaction, constructed
via the characters of the induced Gmaction on spaces of pluricanonical forms. The second
method involves intersection theory on the variety of stable limits of a singular curve. We
compute the expected values for large classes of singular curves, including curves with
ADE, toric and monomial unibranch Gorenstein singularities as well as for ribbons, and show
that the two methods yield identical predictions. We use these results to give a conjectural
outline of the log MMP for Mg.
1. INTRODUCTION
In an effort to understand the canonical model of Mg, Hassett and Keel initiated a program
to give modular interpretations to the log canonical models
Mg() := Proj
d0
