Pruned double Hurwitz numbers

TL;DR

This paper introduces pruned double Hurwitz numbers, a simplified subset of classical double Hurwitz numbers, which retain key structural properties and enable more efficient computations of ramified covers.

Contribution

The paper defines pruned double Hurwitz numbers, proves they satisfy a cut-and-join recursion and are piecewise polynomial, and shows they can be used to compute classical double Hurwitz numbers.

Findings

Pruned double Hurwitz numbers satisfy a cut-and-join recursion.

They are piecewise polynomial in the ramification profiles.

They enable more efficient computation of classical double Hurwitz numbers.

Abstract

Hurwitz numbers count ramified genus $g$, degree $d$ coverings of the projective line with with fixed branch locus and fixed ramification data. Double Hurwitz numbers count such covers, where we fix two special profiles over $0$ and $\infty$ and only simple ramification else. These objects feature insteresting structural behaviour and connections to geometry. In this paper, we introduce the notion of pruned double Hurwitz numbers, generalizing the notion of pruned simple Hurwitz numbers in \cite{DN13}. We show that pruned double Hurwitz numbers, similar to usual double Hurwitz numbers, satisfy a cut-and-join recursion and are piecewise polynomial with respect to the entries of the two special ramification profiles. Furthermore double Hurwitz numbers can be computed from pruned double Hurwitz numbers. To sum up, it can be said that pruned double Hurwitz numbers count a relevant subset of covers, leading to considerably smaller numbers and computations, but still featuring the important properties we can observe for double Hurwitz numbers.