Riemann-Hurwitz formula

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In algebraic geometry The Riemann-Hurwitz formula states that if C,D are smooth algebraic curves, and is a finite map of degree then the number of branch points of , denote by , 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 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 and compute the Euler characteritics of both curves.