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

For the existence of a branched covering $\tilde{Σ} \to Σ$ between closed surfaces there are easy necessary conditions in terms of $χ (\tilde{Σ})$ , $χ (Σ)$ , 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 $Σ$ 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 $\tilde{Σ}$ 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.

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Keywords

surface, branched covering, Riemann-Hurwitz formula

Mathematical Subject Classification 2000

Primary: 57M12

Secondary: 57M30, 57N05

References

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Publication

Received: 20 January 2006

Revised: 14 September 2006

Accepted: 25 September 2006

Published: 14 November 2006

Authors

Ekaterina Pervova
Chelyabinsk State University ul. Br. Kashirinykh, 129 454021 Chelyabinsk, Russia

Carlo Petronio
Dipartimento di Matematica Applicata Università di Pisa Largo Bruno Pontecorvo, 1 56127 Pisa, Italy

Volume 6, issue 4 (2006)

Ekaterina Pervova and Carlo Petronio

Abstract