On the rigidity of moduli of curves in arbitrary characteristic

The stack Mg,nof stable curves and its coarse moduli space Mg,nare defined over Z, and therefore over any field. Over an algebraically closed field of characteristic zero, in [11] Hacking showed that Mg,nis rigid (a conjecture of Kapranov), in [2, 23] Bruno and Mella for g = 0, and the second author for g ≥ 1 showed that its automorphism group is the symmetric group Sn, permuting marked points unless (g,n) ∈ (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 Mg,n, with the same exceptions as over C, implies that its automorphism group is Sn. We prove that, over any field, M0,nis rigid and deduce that, over any field, Aut(M0,n)= Sn for n ≥ 5. Going back to characteristic zero, we prove that for g+n > 4, the coarse moduli spaceMg,nis rigid, extending a result of Hacking who had proven it has no locally trivial deformations. Finally, we show that M1,2 is not rigid, although it does not admit locally trivial deformations, by explicitly computing his Kuranishi family.