On the rigidity of moduli of curves in arbitrary characteristic

The stack $\overline{\mathcal{M}}_{g,n}$ of stable curves and its coarse moduli space $\overline{M}_{g,n}$ are defined over $\mathbb{Z}$, and therefore over any field. Over an algebraically closed field of characteristic zero, Hacking showed that $\overline{\mathcal{M}}_{g,n}$ is rigid (a conjecture of Kapranov). Bruno and Mella for $g=0$, and the second author for $g\geq 1$ showed that its automorphism group is the symmetric group $S_n$, permuting marked points unless $(g,n)\in\{(0,4),(1,1),(1,2)\}$. The methods used in the papers above do not extend to positive characteristic. We show that in characteristic $p>0$, the rigidity of $\overline{\mathcal{M}}_{g,n}$, with the same exceptions as over $\mathbb{C}$, implies that its automorphism group is $S_n$. We prove that, over any perfect field, $\overline{M}_{0,n}$ is rigid and deduce that, over any field, $Aut(\overline{M}_{0,n})\cong S_{n}$ for $n\geq 5$. Going back to characteristic zero, we prove that for $g+n>4$, the coarse moduli space $\overline M_{g,n}$ is rigid, extending a result of Hacking who had proven it has no locally trivial deformations. Finally, we show that $\overline{M}_{1,2}$ is not rigid, although it does not admit locally trivial deformations, by explicitly computing his Kuranishi family.