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

Abstract

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 ${\bar{M}}_g$.

Revised version. 28 pages, 3 figures

References in corpus (4)

• Log minimal model program for the moduli space of stable curves: The first flip
• Weakly proper moduli stacks of curves
• Ample divisors on moduli spaces of weighted pointed rational curves, with applications to log MMP for ${\bar{M}}_{0,n}$
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Cited by in corpus (12)

• Alternate Compactifications of Moduli Spaces of Curves
• Finite Hilbert stability of (bi)canonical curves
• Log minimal model program for the moduli space of stable curves: The second flip
• GIT Constructions of Log Canonical Models of M_g
• Log canonical models and variation of GIT for genus four canonical curves
• The final log canonical model of the moduli space of stable curves of genus four
• GIT and moduli with a twist
• Stability of genus five canonical curves
• An outline of the log minimal model program for the moduli space of curves
• Generalizing the GAGA Principle
• Mori's program for ${\bar{M}}_{0,6}$ with symmetric divisors
• A State Polytope Decomposition Formula