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Surface Branched Covers and Geometric 2-orbifolds

Maria Antonietta Pascali and Carlo Petronio
Transactions of the American Mathematical Society
Vol. 361, No. 11 (Nov., 2009), pp. 5885-5920
Stable URL: http://www.jstor.org/stable/40302941
Page Count: 36
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Surface Branched Covers and Geometric 2-orbifolds
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Abstract

Let ${\tilde \sum }$ and∑ be closed, connected, and orientable surfaces, and let /: $f:\tilde \sum \to \sum $ be a branched cover. For each branching point x ∈ ∑ 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 $X(\tilde \sum ),X(\sum )$ d an d 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 S is not the 2-sphere 5, while for ∑ = 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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