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

Abstract

In this paper, we initiate our investigation of log canonical models for ( M ¯ g , α δ ) (\overline {\mathcal {M}}_g,\alpha \delta ) as we decrease α \alpha from 1 to 0. We prove that for the first critical value α = 9 / 11 \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 α = 7 / 10 \alpha = 7/10 is the next critical value, i.e., the log canonical model stays the same in the interval ( 7 / 10 , 9 / 11 ] (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.