# Double Hurwitz numbers and multisingularity loci in genus 0

## TL;DR

This paper investigates the structure of multisingularity loci in genus 0 Hurwitz spaces, establishing a recursion relation and differential equation that advance understanding of ramification profiles in rational functions.

## Contribution

It introduces a new recursion relation for cohomology classes of multisingularity loci and derives a differential equation governing Hurwitz numbers.

## Key findings

- Established a recursion relation for multisingularity loci cohomology classes
- Derived a differential equation for Hurwitz numbers
- Enhanced understanding of ramification profiles in genus 0

## Abstract

In the Hurwitz space of rational functions on CP^1 with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincare dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.