Moduli space of admissible covers

Admissible covers were developed by Harris and Morrison. The space of admissible covers is a modular compactification of the Hurwitz scheme obtained by allowing the target ${\mathbf{P}}^1$ to degenerate when branch points approach each other. The map ${\mathcal{H}}_{d,g}\to {\mathcal{M}}_g$ given by forgetting the map to ${\mathbf{P}}^1$ extends to a map ${\overline{\mathcal{H}}}_{d,g}\to {\overline{\mathcal{M}}}_g$.

A different compactification of the Hurwitz scheme was given by Abramovich and Vistoli. Their compactification has two advantages: (i) it is smooth, and (ii) it is a moduli space. The disadvantage is that is is a Deligne-Mumford stack rather than a scheme.