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 $\mathcal{H}_{d,g}\rightarrow \mathcal{M}_{g}$ given by forgetting the map to $\mathbf{P}^{1}$ extends to a map $\overline{\mathcal{H}}_{d,g} \rightarrow \overline{\mathcal{M}}_{g}$.

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 ratehr than a scheme.