On double Hurwitz numbers in genus 0

Details On double Hurwitz numbers in genus 0 On double Hurwitz numbers in genus 0

2006-11-14

arxiv.org/abs/math/0611442v1

Mathematics

Algebraic Geometry

17 pages, 3 figures

Scientific paper

We study double Hurwitz numbers in genus zero counting the number of covers $\CP^1\to\CP^1$ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.

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