Boundary behaviour of Hurwitz schemes

Abstract

Actions of finite groups on stable curves are studied. They appear naturally at the boundary of a moduli space of smooth curves with group actions. Those actions which can equivariantly smoothed are characterised. A description of topological types of those actions in terms of the quotient curve and mappings from a kind of fundamental group to the given finite group is given, analogous to the well known case of actions on smooth curves.