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Singularities with 𝔾m-action and\break the log minimal model program for ℳ¯g

  • Jarod Alper EMAIL logo , Maksym Fedorchuk and David Ishii Smyth

Abstract

We give a precise formulation of the modularity principle for the log canonical models M¯g(α):=Projd0H0(¯g,d(K¯g+αδ)) of the moduli space of stable curves. We define a new invariant of Gorenstein curve singularities with 𝔾m-action which can be used to predict the critical α-value at which a singularity first arises in the modular interpretation of M¯g(α). We compute these critical α-values for large classes of singularities with 𝔾m-action, including all ADE, toric planar, and unibranch Gorenstein singularities, and use these results to give a conjectural outline of the log MMP for ¯g.

Award Identifier / Grant number: Postdoctoral Research Fellowship 0802921

Award Identifier / Grant number: DMS-1259226

Award Identifier / Grant number: DMS-0901095

Funding statement: The first author was partially supported by an NSF Postdoctoral Research Fellowship under Grant No. 0802921. The second author was partially supported by NSF grant DMS-1259226. The third author was partially supported by NSF grant DMS-0901095.

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Received: 2011-6-6
Revised: 2014-2-24
Published Online: 2014-8-19
Published in Print: 2016-12-1

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