Definition
Double Hurwitz numbers count the number of covers \(f:C\to \mathbb P^1\) which are branched in a prescribed way at \(0 \) and \(\infty\) and are branched simply elsewhere. More specifically, let \(\alpha = (\alpha_1,...,\alpha_m)\) and \(\beta = (\beta_1,...,\beta_n)\) be partitions of the positive integer \(d\), then the double Hurwitz number \(H^g_{\alpha, \beta}\) is the number of maps \[f:C\to \mathbb P^1\] such that
- \(f \) is degree \(d\) (meaning that generically a fiber has \(d\) points)
- \(f\) has branching given by \(\alpha \) and \(\beta \) at \(0 \) and \(\infty\) respectively
- \(f\) has \(r = 2g - 2 + m + n\) other simply branched points, as determined by Riemann-Hurwitz.
This is of course connected to the Hurwitz scheme