Riemann-Hurwitz formula

For a degree-$d$ holomorphic map, $2g_X-2=d(2g_Y-2)+\sum (e_p-1)$.

For a nonconstant holomorphic map of Riemann surfaces, choose local coordinates centred at $p$ and $f(p)$. The map has the formThe positive integer $e_p$ is its ramification index. The point is ramified exactly when $e_p>1$ and contributes $e_p-1$ to the ramification divisor.

A point $p$ is ramified when its local degree $e_p$ is greater than one.

A branch value is the image of a ramification point of a holomorphic map. Away from all branch values, a proper holomorphic map is an ordinary unramified covering.

Triangulate the target with all branch values among its vertices. For a degree-$d$ map, every edge and face has $d$ lifts. Above a target vertex $v$, the number of lifted vertices isbecause ${\sum }_{p\mapsto v}{e}_p=d$. Thuswhich is equivalent to the Riemann-Hurwitz formula.

Up to invertible affine changes in source and target, every complex cubic polynomial is exactly one ofThe extension to the Riemann sphere has two finite units of ramification. A single finite critical point has ramification index three and gives the first form; two distinct simple critical points can be moved to $\pm 1$, after which integration gives the second.