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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Surface branched covers and geometric 2-orbifolds


Authors: Maria Antonietta Pascali and Carlo Petronio
Journal: Trans. Amer. Math. Soc. 361 (2009), 5885-5920
MSC (2000): Primary 57M12; Secondary 57M50
Published electronically: June 17, 2009
MathSciNet review: 2529918
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Abstract: Let $ \widetilde\Sigma$ and $ \Sigma$ be closed, connected, and orientable surfaces, and let $ f:\widetilde\Sigma\to\Sigma$ be a branched cover. For each branching point $ x\in\Sigma$ the set of local degrees of $ f$ at $ f^{-1}(x)$ is a partition of the total degree $ d$. The total length of the various partitions is determined by $ \chi(\widetilde\Sigma)$, $ \chi(\Sigma)$, $ d$ and the number of branching points via the Riemann-Hurwitz formula. A very old problem asks whether a collection of partitions of $ d$ having the appropriate total length (that we call a candidate cover) always comes from some branched cover. The answer is known to be in the affirmative whenever $ \Sigma$ is not the $ 2$-sphere $ S$, while for $ \Sigma=S$ exceptions do occur. A long-standing conjecture however asserts that when the degree $ d$ is a prime number a candidate cover is always realizable. In this paper we analyze the question from the point of view of the geometry of 2-orbifolds, and we provide strong supporting evidence for the conjecture. In particular, we exhibit three different sequences of candidate covers, indexed by their degree, such that for each sequence:

  • The degrees giving realizable covers have asymptotically zero density in the naturals.
  • Each prime degree gives a realizable cover.


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Additional Information

Maria Antonietta Pascali
Affiliation: Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro, 2, 00185 Roma, Italy
Email: pascali@mat.uniroma1.it

Carlo Petronio
Affiliation: Dipartimento di Matematica Applicata, Università di Pisa, Via Filippo Buonarroti, 1C, 56127 Pisa, Italy
Email: petronio@dm.unipi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04779-5
PII: S 0002-9947(09)04779-5
Received by editor(s): September 17, 2007
Published electronically: June 17, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.