Riemann–Hurwitz formula

The formula may also be used to calculate the genus of hyperelliptic curves.

Firstly, there are no ramified covering maps from a curve of lower genus to a curve of higher genus – and thus, since non-constant meromorphic maps of curves are ramified covering spaces, there are no non-constant meromorphic maps from a curve of lower genus to a curve of higher genus.

As another example, it shows immediately that a curve of genus 0 has no cover with N > 1 that is unramified everywhere: because that would give rise to an Euler characteristic > 2.

An orbifold covering of degree N between orbifold surfaces S' and S is a branched covering, so the Riemann–Hurwitz formula implies the usual formula for coverings