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 $\pi: C \rightarrow \mathbf{P}^{1}$ with simple ramification. Variations on this definition include allowing more complicated ramification behavior and higher genus targets.