Log canonical models for the moduli space of curves: The first divisorial contraction

Hassett, Brendan; Hyeon, Donghoon

doi:10.1090/S0002-9947-09-04819-3

Log canonical models for the moduli space of curves: The first divisorial contraction
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Trans. Amer. Math. Soc. 361 (2009), 4471-4489 Request permission

Abstract:

In this paper, we initiate our investigation of log canonical models for $(\overline {\mathcal {M}}_g,\alpha \delta )$ as we decrease $\alpha$ from 1 to 0. We prove that for the first critical value $\alpha = 9/11$, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that $\alpha = 7/10$ is the next critical value, i.e., the log canonical model stays the same in the interval $(7/10, 9/11]$. In the appendix, we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space.

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