Compactifications of Hurwitz Spaces

We construct several modular compactifications of the Hurwitz space H-g/h(d) of genus g curves expressed as d-sheeted, simply branched covers of genus h curves. These compactifications are obtained by allowing the branch points of the covers to collide to a variable extent. They are well behaved if d=2,3, or if relatively few collisions are allowed. We recover as special cases the spaces of twisted admissible covers of Abramovich, Corti, and Vistoli and the spaces of hyperelliptic curves of Fedorchuk.