Explicit computation of some families of Hurwitz numbers

TL;DR

This paper explicitly computes certain Hurwitz numbers, counting branched covers with specified local degrees, using combinatorial methods based on dessins d'enfant, providing formulas for various genus and branching configurations.

Contribution

It introduces explicit arithmetic formulas for specific families of Hurwitz numbers using a combinatorial approach with dessins d'enfant.

Findings

Derived explicit formulas for Hurwitz numbers with given local degrees.

Extended computations to various genus g and parameter h.

Demonstrated the effectiveness of dessins d'enfant in enumerating branched covers.

Abstract

We compute the number of (weak) equivalence classes of branched covers from a surface of genus g to the sphere, with 3 branching points, degree 2k, and local degrees over the branching points of the form (2,...,2), (2h+1,1,2,...,2), (d_1,...,d_m), for several values of g and h. We obtain explicit formulae of arithmetic nature in terms of the d_i's. Our proofs employ a combinatorial method based on Grothendieck's dessins d'enfant.