On the existence of branched coverings between surfaces with prescribed branch data, I

By E. Pervova, Carlo Petronio. 2005

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For the existence of a branched covering Sigma~ --> Sigma between closed surfaces there are easy necessary conditions in terms of chi(Sigma~), chi(Sigma), orientability, the total degree, and the local degrees at the branching points. A classical problem dating back to Hurwitz asks whether these conditions are also sufficient. Thanks to the work of many authors, the problem remains open only when Sigma is the sphere, in which case exceptions to existence are known to occur. In this paper we describe new infinite series of exceptions, in particular previously unknown exceptions with Sigma~ not the sphere and with more than three branching points. All our series come with systematic explanations, based on several different techniques (including dessins d'enfants and decomposability) that we exploit to attack the problem, besides Hurwitz's classical technique based on permutations. Using decomposability we also establish an easy existence result.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8fa00ce5c70dd9e6e14d06b878a9c249e26ae2e4

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In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Gonality of algebraic curves and graphs (2012, background): “ = (2,1,...,1) for every i, or d ≤ 3, or b ≤ 2. On the other hand if d = 4 the partition set P = {(3,1);(2,2);(2,2)} is not of Hurwitz type, but the Riemann-Hurwitz formula (16) holds with g = 0; see [PP] for this and other results on the Hurwitz existence problem. Let now φ : (G,w) → T be a non-deg”
Lifting harmonic morphisms II: tropical curves and metrized complexes (2014, methodology): “We refer the reader to [Edmonds et al. 1984; Pervova and Petronio 2006; 2008], along with the references therein, for an account of what is known about this subject.”
Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta (2013, background): “nishing Hurwitz numbers are known (see Example 8.5). However, in general one has to explicitly compute a given Hurwitz number to decide if this latter vanishes or not. We refer the reader to [EKS84], [PP06], and [PP08], along with the references therein, for an account of what is known about this su”