Singularities with G_m-action and the log minimal model program for $\bar{M}_g$

We give a precise formulation of the modularity principle for the log canonical models of $\bar{M}_g$. Assuming the modularity principle holds, we develop and compare two methods for determining the critical alpha-values at which a singularity or complete curve with G_m-action arises in the modular interpretations of log canonical models of $\bar{M}_g$. The first method involves a new invariant of curve singularities with G_m-action, constructed via the characters of the induced G_m-action 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 alpha-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 $\bar{M}_g$.