Counting ramified coverings and intersection theory on spaces of rational functions I (Cohomology of Hurwitz spaces)

摘要

The Hurwitz space is a compactification of the space of rational functions ofa given degree. The Lyashko-Looijenga map assigns to a rational function theset of its critical values. It is known that the number of ramified coveringsof CP^1 by CP^1 with prescribed ramification points and ramification types isrelated to the degree of the Lyashko--Looijenga map on various strata of theHurwitz space. Here we explain how the degree of the Lyashko-Looijenga map isrelated to the intersection theory on this space. We describe the cohomologyalgebra of the Hurwitz space and prove several relations between the homologyclasses represented by various strata.