The final log canonical model of $\overline{\mathcal{M}}_6$

Abstract

We describe the birational model of ${\overline{\mathcal{ℳ}}}_6$ given by quadric hyperplane sections of the degree-$5$ del Pezzo surface. In the spirit of the genus-$4$ case treated by Fedorchuk, we show that it is the last nontrivial space in the log minimal model program for ${\overline{\mathcal{ℳ}}}_6$. We also obtain a new upper bound for the moving slope of the moduli space.