2014/10/23 by Enrica Duchi, Duchi, Enrica, Dominique Poulalhon +3
Mathematics · #05C30 #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1410.6521
openalex publication_date 2014/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a bijective proof of Hurwitz formula for the number of simple branched coverings of the sphere by itself. Our approach extends to double Hurwitz numbers and yields new properties for them. In particular we prove for double Hurwitz numbers a conjecture of Kazarian and Zvonkine, and we give an expression that in a sense interpolates between two celebrated polynomiality properties: polynomiality in chambers for double Hurwitz numbers, and a new analog for almost simple genus 0 Hurwitz numbers of the polynomiality up to normalization of simple Hurwitz numbers of genus g. Some probabilistic implications of our results for random branched coverings are briefly discussed in conclusion.