Double Hurwitz numbers and multisingularity loci in genus 0

In the Hurwitz space of rational functions on the complex projective line 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 Poincaré dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers