  To a branched cover between closed, connected and orientable surfaces one
associates a "branch datum", which consists of the two surfaces, the total
degree d, and the partitions of d given by the collections of local degrees
over the branching points. This datum must satisfy the Riemann-Hurwitz formula.
A "candidate surface cover" is an abstract branch datum, a priori not coming
from a branched cover, but satisfying the Riemann-Hurwitz formula. The old
Hurwitz problem asks which candidate surface covers are realizable by branched
covers. It is now known that all candidate covers are realizable when the
candidate covered surface has positive genus, but not all are when it is the
2-sphere. However a long-standing conjecture asserts that candidate covers with
prime degree are realizable. To a candidate surface cover one can associate one
Y -> X between 2-orbifolds, and in a previous paper we have completely analyzed
the candidate surface covers such that either X is bad, spherical, or
Euclidean, or both X and Y are rigid hyperbolic orbifolds, thus also providing
strong supporting evidence for the prime-degree conjecture. In this paper,
using a variety of different techniques, we continue this analysis, carrying it
out completely for the case where X is hyperbolic and rigid and Y has a
2-dimensional Teichmueller space. We find many more realizable and
non-realizable candidate covers, providing more support for the prime-degree
conjecture.
