On double Hurwitz numbers in genus 0

TL;DR

This paper investigates the piecewise polynomial structure of genus zero double Hurwitz numbers, describing the chamber decomposition, differences between polynomials in neighboring chambers, and providing explicit formulas for calculations.

Contribution

It characterizes the chamber structure of double Hurwitz numbers in genus zero and derives explicit formulas for the polynomials in specific chambers, enabling systematic computation.

Findings

Partition of parameter space into polynomiality chambers

Explicit difference formulas between neighboring chambers

Polynomial formula for the totally negative chamber

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.