Riemann-Hurwitz formula

In algebraic geometry The Riemann-Hurwitz formula states that if C,D are smooth algebraic curves, and $f:C\to D$ is a finite map of degree $d$ then the number of branch points of $f$, denote by $B$, is given by

$2(genus(C)-1)=2d(genus(D)-1)+B$.

Over a field in general characteristic, this theorem is a consequence of the Riemann-Roch theorem. Over the complex numbers, the theorem can be proved by choosing a triangulation of the curve $D$ such that all the branch points of the map are nodes of the tringulation. One then consider the pullback of the tringulation to the curve $C$ and compute the Euler characteritics of both curves.