Hurwitz schemes

The Hurwitz scheme ${\mathcal{H}}_{d,g}$ parametrizes simply branched covers of ${\mathbf{P}}^1$, i.e. pairs $(C,\pi )$ of a genus $g$ curve $C$ and a degree $d$ map $\pi :C\to {\mathbf{P}}^1$ with simple ramification. Variations on this definition include allowing more complicated ramification behavior and higher genus targets.