For the existence of a branched covering
between closed surfaces there are easy necessary conditions in terms of
,
,
orientability, the total degree, and the local degrees at the branching points. A
classical problem dating back to Hurwitz asks whether these conditions are also
sufficient. Thanks to the work of many authors, the problem remains open only when
is the
sphere, in which case exceptions to existence are known to occur. In this paper we
describe new infinite series of exceptions, in particular previously unknown exceptions
with
not the sphere and with more than three branching points. All our series come with
systematic explanations, based on several different techniques (including dessins
d’enfants and decomposability) that we exploit to attack the problem, besides
Hurwitz’s classical technique based on permutations. Using decomposability we also
establish an easy existence result.
Keywords
surface, branched covering, Riemann-Hurwitz formula
