On double Hurwitz numbers in genus 0

Abstract

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.

17 pages, 3 figures

References in corpus (8)

• Hurwitz numbers and intersections on moduli spaces of curves
• On Hurwitz numbers and Hodge integrals
• Hodge integrals and Hurwitz numbers via virtual localization
• Geometry of meromorphic functions and intersections on moduli spaces of curves
• Topological recursive relations in $H^{2g}(M_{g,n})$
• Towards the Intersection Theory on Hurwitz Spaces
• Multipoint series of Gromov-Witten invariants of CP^1
• Counting ramified coverings and intersection theory on Hurwitz spaces II (Local structure of Hurwitz spaces and combinatorial results)

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• Double Hurwitz numbers: polynomiality, topological recursion and intersection theory
• Double Hurwitz numbers via the infinite wedge
• The Universality of the Resonance Arrangement and its Betti Numbers
• The combinatorics of real double Hurwitz numbers with real positive branch points
• Bijections for simple and double Hurwitz numbers
• Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics
• Tropical curves and covers and their moduli spaces
• Chamber structure for some equivariant relative Gromov-Witten invariants of ${\mathbb{P}}^1$ in genus $0$
• Completed Cycles Leaky Hurwitz Numbers
• On the lower bounds for real double Hurwitz numbers