# Riemann-Hurwitz formula

From Knowino

In algebraic geometry the **Riemann-Hurwitz formula**, named after Bernhard Riemann and Adolf Hurwitz, states that if *C*, *D* are smooth algebraic curves, and is a finite map of degree *d* then the number of branch points of *f*, denoted by *B*, is given by

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 triangulation. One then considers the pullback of the triangulation to the curve *C* and computes the Euler characteristics of both curves.

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