Let and be closed, connected, and orientable surfaces, and let
be
a branched cover. For each branching
point the set of
local degrees of at
is a partition of the total degree
. The
total length of the various partitions
is determined by , ,
and
the number of branching points
via the Riemann-Hurwitz formula. A very old problem asks
whether a collection of partitions of
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 is not the
-sphere
, while
for exceptions do occur. A long-standing conjecture however asserts that when the
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:
- The
degrees giving
realizable covers have
asymptotically zero density in the naturals.
- Each
prime degree gives a
realizable cover.