Compactifications of Hurwitz spaces

We construct several modular compactifications of the Hurwitz space $H^d_{g/h}$ 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 very 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.