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In this note we provide a new partial solution to the Hurwitz existenceproblem for surface branched covers. Namely, we consider candidate branch datawith base surface the sphere and one partition of the degree having length two,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 somewhatrecently by Pakovich, and we deal here with the case of positive genus. We showthat the only other exceptional candidate data, besides those of Pakovich (fiveinfinite families and one sporadic case), are a well-known very specificinfinite family in degree 4 (indexed by the genus of the candidate coveringsurface, which can attain any value), five sporadic cases (four in genus 1 andone 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, ourfindings provide more evidence for the prime-degree conjecture. Our argumentproceeds by induction on the genus and on the number of branching points, soour results logically depend on those of Pakovich, and we do not employ thetechnology of constellations on which his proof is based.

Solution of the Hurwitz problem with a length-2 partition