Riemann-Hurwitz formula

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

[math]\displaystyle{ 2(genus(C)-1)=2d(genus(D)-1)+B }[/math].

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 [math]\displaystyle{ D }[/math] 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 [math]\displaystyle{ C }[/math] and compute the Euler characteritics of both curves.