Solution of the Hurwitz problem with a length-2 partition

Abstract

In this note, we provide a new partial solution to the Hurwitz existence problem for surface branched covers. Namely, we consider candidate branch data with base surface the sphere and one partition of the degree having length 2, and we fully determine which of them are realizable and which are exceptional. The case where the covering surface is also the sphere was solved somewhat recently by Pakovich, and we deal here with the case of positive genus. We show that the only other exceptional candidate data, besides those of Pakovich (five infinite families and one sporadic case), are a well-known, very specific infinite family in degree 4 (indexed by the genus of the candidate covering surface, which can attain any value), five sporadic cases (four in genus 1 and one in genus 2), and another infinite family in genus 1 also already known. Since the degree is a composite number for all these exceptional data, our findings provide more evidence for the prime-degree conjecture. Our argument proceeds by induction on the genus and on the number of branching points, so our results logically depend on those of Pakovich, and we do not employ the technology of constellations on which his proof is based.