Singularities with \(\mathbb{G}_m\)-action and the log minimal model program for \(\overline{\mathcal{M}}_g\). (English) Zbl 1354.14041
The log minimal model program (log MMP) on the moduli space \(\overline{\mathcal{M}}_g\) of stable genus \(g\) curves (also known as the Hassett-Keel program) can produce new birational models of \(\overline{\mathcal{M}}_g\) with modular meanings, i.e. parameterizing curves with various singularities. In this very interesting paper, the authors give a precise formulation of this modularity principle. They also define a new invariant of Gorenstein curve singularities with \(\mathbb G_m\)-action and use it to predict at which stage a singularity first arises in the log MMP. They work out a number of examples, including all ADE, toric planar, and unibranch Gorenstein singularities, and use these results to give a conjectural outline of the log MMP for \(\overline{\mathcal{M}}_g\).
Reviewer: Dawei Chen (Chestnut Hill)
MSC:
| 14H10 | Families, moduli of curves (algebraic) |
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
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