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Rational functions with given ramification in characteristic p

Published online by Cambridge University Press:  13 March 2006

Brian Osserman
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, [email protected]
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Abstract

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Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of rational functions (up to automorphism of the target) on the projective line with ramification to order ei at general points Pi, in the case that all ei are less than the characteristic. Unlike the case of characteristic 0, the answer is not given by Schubert calculus, nor is the number of maps always finite for distinct Pi, even in the tamely ramified case. However, finiteness for general Pi, obtained by exploiting the relationship to branched covers, is a key part of the argument.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2006