N ov 2 00 6 On the existence of branched coverings between surfaces with prescribed branch data , II

If Σ̃ → Σ is a branched covering between closed surfaces, there are several easy relations one can establish between χ(Σ̃), χ(Σ), orientability of Σ and Σ̃, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary relations have been shown to be also sufficient for the existence of the covering except when Σ is the sphere S (and when Σ is the projective plane, but this case reduces to the case Σ = S). For Σ = S many exceptions are known to occur and the problem is widely open. Generalizing methods of Baránski, we prove in this paper that the necessary relations are actually sufficient in a specific but rather interesting situation. Namely under the assumption that Σ = S, that there are three branching points, that one of these branching points has only two preimages with one being a double point, and either that Σ̃ = S and that the degree is odd, or that Σ̃ has genus at least one, with a single specific exception. For the case of Σ̃ = S we also show that for each even degree there are precisely two exceptions. MSC (2000): 57M12. ∗Partially supported by the INTAS YS fellowship 03-55-1423 ∗∗Supported by the INTAS project “CalcoMet-GT” 03-51-3663