On the existence of branched coverings between surfaces with prescribed branch data

Given a branched covering between closed connected surfaces, one can easily establish some relations between the Euler characteristic and orientability of the involved surfaces, the degree of the covering, the number of branching points, and the local degrees at these points. These relations can therefore be regarded as necessary conditions for the existence of the covering. A classical problem dating back to Hurwitz asks whether these necessary conditions are actually also sufficient. Thanks to the work of many authors, the problem is now completely solved (in the affirmative) when the base surface has non-positive Euler characteristic. The cases where the base surface is the sphere or the projective plane remain elusive, but the latter reduces to the former, and many partial results are known. In this paper we prove several new existence and non-existence results. MSC (2000): 57M12.