Solution of the Hurwitz problem with a length-2 partition

In this note, we provide a new partial solution to the Hurwitz existence problemfor surface branched covers. Namely, we consider candidate branch data with base surfacethe sphere and one partition of the degree having length2, and we fully determine whichof them are realizable and which are exceptional. The case where the covering surface isalso the sphere was solved somewhat recently by Pakovich, and we deal here with the caseof positive genus. We show that the only other exceptional candidate data, besides thoseof Pakovich (five infinite families and one sporadic case), are a well-known, very specificinfinite family in degree4(indexed by the genus of the candidate covering surface, whichcan attain any value), five sporadic cases (four in genus1and one in genus2), and anotherinfinite family in genus1also already known. Since the degree is a composite number for allthese 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 technologyof constellations on which his proof is based