An Outline of the Log Minimal Model Program for the Moduli Space of Curves

ABSTRACT

We consider the log minimal model program for the moduli space of stable curves. It is widely believed now that there is a descending sequence of critical $α$ values where the log canonical model for the moduli space of stable curves with respect to $αδ$ changes, where $δ$ denotes the divisor of singular curves. We derive a conjectural formula for the critical values in two different ways: By working out the intersection theory of the moduli space of hyperelliptic curves and by computing the GIT stability of certain curves with tails and bridges. The results give a rough outline of how the log minimal model program would proceed, predicting when the log canonical model changes and which curves are to be discarded and acquired at the critical steps.

KEYWORDS:

• moduli of curves
• log minimal model program
• Geometric Invariant Theory

2010 AMS Subject Classification:

• 14D22; 14E30

Acknowledgments

The author thanks Brendan Hassett for sharing his ideas on the log minimal model program for the moduli of stable curves. Especially, the results in Section 2.1 were developed from his stability computation of genus two tails. The author is very grateful to the anonymous referee who took pains to review the article in great detail and gave numerous suggestions which greatly improved the paper. The author thanks Sean Keel for pointing out inaccuracies in a preliminary version, and Jarod Alper and Maksym Fedorchuk for patiently explaining their heuristics of predicting critical values and change in singularity loci. He also greatly benefited from useful conversations with Valery Alexeev, Angela Gibney, Young-Hoon Kiem, Yongnam Lee, Ian Morrison, Jihun Park, David Smyth, and Dave Swinarski.

Notes

1 Up to the point when δ_i becomes general type.

2 This is a natural generalization of the picture described in [Hassett and Hyeon 09, Hyeon and Lee 10b] in which the divisorial contraction (resp. ) replaces an elliptic tail with an ordinary cusp.