Recobrimentos ramificados entre superfícies e 2-orbifolds geométricos

Abstract

The purpose of this work is to study the realizability problem of branched coverings between closed, connected and orientable surfaces. For each covering, there exists a set of naturally associated data called branch datum that should satisfy the Riemann-Hurwitz formula. A classical problem (for possibly non-orientable surfaces) asks whether for a branch datum satisfying the condition of Riemann-Hurwitz exists a branched cover between surfaces having it as branch datum. The correct answer is: not always. When a branch datum satisfies the necessary conditions to come from a branched covering, we call it a candidate branched covers ; if indeed it comes from a branched cover we call it realizable and, if not, we call it exceptional. In fact, it is known that exceptions can occur only if the covered surface is the sphere or the projective plane, but the general solution is still unknown. Among the various tools used to attack the problem we will work directly with two of them: the orbifolds and dessins d'enfant.