Branched covers between surfaces

Let e and be closed, connected, and orientable surfaces, and let f: e ! be a branched cover. For each branching point x 2 the set of local degrees of f at f1(x) is a partition of the total degree d. The total length of the various partitions is determined by (e), (), 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 armative whenever is not the 2-sphere S, 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 the main result of this thesis, proved in Chapter 3, we provide strong supporting evidence for the conjecture. In particular, we exhibit three dierent sequences of candidate covers, indexed by their degree, such that for each sequence