Riemann-Hurwitz formula: Difference between revisions
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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>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>C</math> and compute the [[Euler characteritics]] of both curves. | 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>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>C</math> and compute the [[Euler characteritics]] of both curves. | ||
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Revision as of 00:10, 16 February 2007
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.
