  For a branched cover between two closed orientable surfaces, the
Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the
total degree of the cover, and the total length of the partitions of the degree
given by the local degrees at the preimages of the branching points. A very old
problem asks whether a collection of partitions of an integer 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 the
candidate base surface is not the 2-sphere, while for the 2-sphere exceptions
do occur. A long-standing conjecture however asserts that when the candidate
degree 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: (1) The degrees giving realizable covers have
asymptotically zero density in the naturals; (2) Each prime degree gives a
realizable cover.
