Degrees of cohomological classes of multisinguliarities in Hurwitz spaces of rational functions

TL;DR

This paper develops new cohomological formulas for degrees of specific strata in Hurwitz spaces of rational functions with two degenerate critical values, leading to explicit formulas for certain double Hurwitz numbers.

Contribution

It introduces new cohomological relations and explicit formulas for degrees of strata in Hurwitz spaces, extending previous conjectures and computational evidence.

Findings

Derived new formulas for degrees of strata with one codimension-1 degeneracy.

Proved conjectured cohomological relations in Hurwitz spaces.

Obtained explicit formulas for certain double Hurwitz numbers in genus 0.

Abstract

The main goal of the present paper are new formulae for degrees of strata in Hurwitz spaces of rational functions having two degenerate critical values with preimages of prescribed multiplicities. We consider the case where the multiplicities of the preimages of one critical value are arbitrary, while the second critical value has degeneracy of codimension~$1$. Our formulae are based on the universal cohomological expressions for codimension~$1$ strata in terms of certain basic cohomology classes in general Hurwitz spaces of rational functions obtained by M.~Kazarian and S.~Lando. We prove new relations valid in cohomology of Hurwitz spaces that were conjectured by M.~Kazarian on the base of computer experiments. As a corollary, we obtain new, previously unknown, explicit formulae for certain families of double Hurwitz numbers in genus~$0$. One may hope that the methods developed in the present paper are applicable to proving more general relations in cohomology rings of Hurwitz spaces and deducing more general formulae for double Hurwitz numbers.