Surface branched covers and geometric 2-orbifolds

Abstract

Let $\widetilde \Sigma$ and $\Sigma$ be closed, connected, and orientable surfaces, and let $f:\widetilde \Sigma \to \Sigma$ be a branched cover. For each branching point $x\in \Sigma$ the set of local degrees of $f$ at $f^{-1}(x)$ is a partition of the total degree $d$. The total length of the various partitions is determined by $\chi (\widetilde \Sigma )$, $\chi (\Sigma )$, $d$ and the number of branching points via the Riemann-Hurwitz formula. A very old problem asks whether a collection of partitions of $d$ 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 $\Sigma$ is not the $2$-sphere $S$, while for $\Sigma =S$ exceptions do occur. A long-standing conjecture however asserts that when the degree $d$ 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.