Surface branched covers and geometric 2-orbifolds

Let Σ̄ and Σ be closed, connected, and orientable surfaces, and let f: Σ̄ → Σ be a branched cover. For each branching point x ∈ Σ 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 Χ(Σ̄), Χ(Σ), 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 Σ is not the 2-sphere S, while for Σ = 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. © 2009 American Mathematical Society.