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

Brendan Hassett·Donghoon Hyeon
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Abstract

In this paper, we initiate our investigation of log canonical models for the moduli space of curves with the boundary divisor $\a \d$ as we decrease $\a$ from 1 to 0. We prove that for the first critical value $\a = 9/11$, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that $\a = 7/10$ 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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Brendan Hassett, Donghoon Hyeon. 2006-07-19. Log canonical models for the moduli space of curves: First divisorial contraction. https://arxiv.org/abs/math/0607477

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