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The main goal of the present paper are new formulae for degrees of strata inHurwitz spaces of rational functions having two degenerate critical values withpreimages of prescribed multiplicities. We consider the case where themultiplicities of the preimages of one critical value are arbitrary, while thesecond critical value has degeneracy of codimension~$1$. Our formulae are basedon the universal cohomological expressions for codimension~$1$ strata in termsof certain basic cohomology classes in general Hurwitz spaces of rationalfunctions obtained by M.~Kazarian and S.~Lando. We prove new relations valid incohomology of Hurwitz spaces that were conjectured by M.~Kazarian on the baseof 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 toproving more general relations in cohomology rings of Hurwitz spaces anddeducing more general formulae for double Hurwitz numbers.

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