Explicit computation of some families of Hurwitz numbers, II

TL;DR

This paper extends the combinatorial computation of Hurwitz numbers for specific branched covers involving genus g surfaces, using dessins d'enfant, providing explicit formulas for various cases.

Contribution

It introduces a method to explicitly compute Hurwitz numbers for certain branched covers with genus g source surfaces and three branch points, expanding previous results.

Findings

Derived explicit formulas for Hurwitz numbers in specific cases

Computed values for various genus g and h

Enhanced combinatorial methods for surface branched covers

Abstract

We continue our computation, using a combinatorial method based on Gronthendieck's dessins d'enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surface, 3 branching points, degree 2k, and local degrees over the branching points of the form [2,...,2], [2h+1,3,2,...,2], [d_1,...,d_l]. We compute the corresponding (weak) Hurwitz numbers for several values of g and h, getting explicit arithmetic formulae in terms of the d_i's.