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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Surface branched covers and geometric 2-orbifolds
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by Maria Antonietta Pascali and Carlo Petronio PDF
Trans. Amer. Math. Soc. 361 (2009), 5885-5920 Request permission

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
    • Received by editor(s): September 17, 2007
    • Published electronically: June 17, 2009
    • © Copyright 2009 American Mathematical Society
      The copyright for this article reverts to public domain 28 years after publication.
    • Journal: Trans. Amer. Math. Soc. 361 (2009), 5885-5920
    • MSC (2000): Primary 57M12; Secondary 57M50
    • DOI: https://doi.org/10.1090/S0002-9947-09-04779-5
    • MathSciNet review: 2529918