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

In this paper, we initiate our investigation of log canonical models for $(\overline{\bf\mathcal{M}}_g,\alpha \delta)$ 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.