José Bertin and Ariane Mézard
Let $C$ be a generically smooth, locally complete intersection curve defined over an algebraically closed field $k$ of characteristic $p\geq 0$. Let $G\subset{}$Aut${}_k C$ be a finite group of automorphisms of $C$. We develop a theory of $G$-equivariant deformations of the Galois cover $C\rightarrow C/G$. We prove that all obstructions to $G$-equivariant deformations are local: they are localized at singular and widely ramified points. As an application, we discuss the case of $G$-equivariant deformations of semi-stable curves.