Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/30767
Title: Realizations of certain odd-degree surface branch data
Authors: Petronio, Carlo
Keywords: Surface branched coverHurwitz number
Issue Date: 2020
Publisher: EUT Edizioni Università di Trieste
Source: Carlo Petronio, "Realizations of certain odd-degree surface branch data" in: "Rendiconti dell’Istituto di matematica dell’Università di Trieste: an International Journal of Mathematics vol. 52 (2020)", EUT Edizioni Università di Trieste, Trieste, 2020
Journal: Rendiconti dell’Istituto di matematica dell’Università di Trieste: an International Journal of Mathematics 
Abstract: 
We consider surface branch data with base surface the sphere, odd degree d, three branching points, and partitions of d of the form

(2, ..., 2, 1) (2,..., 2, 2h + 1) π

with π having length ℓ. This datum satisfies the Riemann-Hurwitz necessary condition for realizability if h — ℓ is odd and at least —1. For several small values of h and ℓ (namely, for h + ℓ ≤ 5) we explicitly compute the number v of realizations of the datum up to the equivalence relation given by the action of automorphisms (even unoriented ones) of both the base and the covering surface. The expression of v depends on arithmetic properties of the entries of π. In particular we find that in the only case where v is 0 the entries of π have a common divisor, in agreement with a conjecture of Edmonds-Kulkarny-Stong and a stronger one of Zieve.
Type: Article
URI: http://hdl.handle.net/10077/30767
ISSN: 0049-4704
eISSN: 2464-8728
DOI: 10.13137/2464-8728/30767
Rights: Attribution-NonCommercial-NoDerivatives 4.0 Internazionale
Appears in Collections:Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.52 (2020), in progress

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