DEFORMATION OF CURVES WITH TAME ACTION

Let k be a field, C/k a stable curve of genus g greater-than-or-equal-to 2, and G a finite group with tame action on C/k. We prove that the functor of the G-equivariant deformations of C is formally smooth, and even formally etale for certain curves. A theorem of Nakajima enables us to achieve this by showing that the k[G]-module Ext1(OMEGA(C/k), O(c)) is projective, whence H*-acyclic. A corollary (6.1) is the finiteness of the number of smooth curves of genus g greater-than-or-equal-to 2 with an automorphism group of order > 12(g - 1) over any algebraically closed field of characteristic p, with p = 0 or p > g + 1. Another corollary (5.2) concerns the lifting of pairs (C, G) from characteristic p to characteristic 0. A third corollary (5.3) is a criterion of good reduction for curves with automorphisms.