Log canonical models for the moduli space of curves: first divisorial contraction (following the paper of B. Hasset and D. Hyeon)

We discuss changes of log canonical model $(\overline{\mathrm{M}}_g,\alpha\delta)$ under decreasing $\alpha$ from $1$ to $0$ where $\delta$ is a boundary divisor. We prove that for the first critical value $\alpha=\frac{9}{11}$ log canonical model is isomorphic to a moduli space of semistable curves havштп ordinary cusps and tacnodes as singularities. We show that the next critical value of $\alpha$ is $\frac{7}{10}$. In particular a log canonical model does not change on the interval $(\frac{7}{10},\frac{9}{11}]$.