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We give a precise formulation of the modularity principle for the logcanonical models of $\bar{M}_g$. Assuming the modularity principle holds, wedevelop and compare two methods for determining the critical alpha-values atwhich a singularity or complete curve with G_m-action arises in the modularinterpretations of log canonical models of $\bar{M}_g$. The first methodinvolves a new invariant of curve singularities with G_m-action, constructedvia the characters of the induced G_m-action on spaces of pluricanonical forms.The second method involves intersection theory on the variety of stable limitsof a singular curve. We compute the expected alpha-values for large classes ofsingular curves, including curves with ADE, toric, and monomial unibranchGorenstein singularities, as well as for ribbons, and show that the two methodsyield identical predictions. We use these results to give a conjectural outlineof the log MMP for $\bar{M}_g$.

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