Surface branched covers and geometric 2-orbifolds

TL;DR

This paper investigates the realizability of branched covers between surfaces using 2-orbifold geometry, providing evidence supporting the conjecture that prime degree covers are always realizable, with specific sequences illustrating the phenomenon.

Contribution

It introduces a geometric approach to the classical problem, offering new sequences of candidate covers and supporting the prime degree conjecture through asymptotic and prime-specific results.

Findings

Prime degree candidate covers are always realizable.

Sequences show asymptotically zero density of realizable degrees.

Geometric analysis supports the conjecture for 2-orbifolds.

Abstract

For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whether a collection of partitions of an integer 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 the candidate base surface is not the 2-sphere, while for the 2-sphere exceptions do occur. A long-standing conjecture however asserts that when the candidate 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: (1) The degrees giving realizable covers have asymptotically zero density in the naturals; (2) Each prime degree gives a realizable cover.