Counting ramified coverings and intersection theory on Hurwitz spaces II (Local structure of Hurwitz spaces and combinatorial results)

TL;DR

This paper investigates the structure of Hurwitz spaces, focusing on boundary intersections and cohomology, to derive recurrence relations and generating functions for counting ramified coverings of spheres.

Contribution

It introduces new methods to analyze the boundary structure of Hurwitz spaces and derives recurrence relations for enumerating ramified coverings.

Findings

Recurrence relations for counting ramified coverings

Identification of generating functions within a specific subalgebra

Insights into the cohomology ring of Hurwitz spaces

Abstract

The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.