Deformation of curves with a group action

Abstract

In this thesis we study the deformations of curves with a finite group action over an algebraically closed field of positive characteristic. While the deformation theory of curves is well-known, there is no complete picture when a group action comes into play. When the genus of the curve is greater than 1, it is known that the deformation functor of the curve with a group action is pro-representable. For a genus 0 curve, we prove in this thesis that the deformation functor is non-pro-representable exactly when the field characteristic is 2, and the group is Z/2, (Z/2)^2 or the dihedral group D_n with n odd. Proving pro-representability in the other cases relies on reduction to local deformations. The non-pro-representability is proved by direct calculations. For genus 1 curves, the problem is still open. We propose an approach that might deal with elliptic curves with a small automorphism group.