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Explicit computation of some families of Hurwitz numbers, II

Carlo Petronio
From the journal Advances in Geometry

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], π=[di]i=1. We compute the corresponding (weak) Hurwitz numbers for several values of g and h, getting explicit arithmetic formulae in terms of the di’s.

MSC 2010: 57M12

Funding statement: The author was partially supported by INdAM through GNSAGA, by MIUR through PRIN “Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis” and by UniPI through PRA 2018/22 “Geo-metria e Topologia delle varietà”.

  1. Communicated by: G. Gentili

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Received: 2018-08-07
Published Online: 2020-10-08
Published in Print: 2020-10-27

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