The Hurwitz existence problem for surface branched covers

Petronio, Carlo

Winter Braids X (Pisa, 2020), Winter Braids Lecture Notes (2020), Exposé no. 2, 43 p.

To a branched cover $f:\tilde{\Sigma }\to \Sigma$ between closed surfaces one can associate a combinatorial datum given by the topological types of $\tilde{\Sigma }$ and $\Sigma$, the degree $d$ of $f$, the number $n$ of branching points of $f$, and the $n$ partitions of $d$ given by the local degrees of $f$ at the preimages of the branching points. This datum must satisfy the Riemann-Hurwitz condition plus some extra ones if either $\Sigma$ or both $\Sigma$ and $\tilde{\Sigma }$ are non-orientable. A very old question posed by Hurwitz [14] in 1891 asks whether a combinatorial datum satisfying these necessary conditions is actually realizable (namely, associated to some existing $f$) or not (in which case it is called exceptional). Or, more generally, to count the number of realizations of the datum up to a natural equivalence relation. Many partial answers have been given to the Hurwitz problem over the time, but a complete solution is still missing. In this short course we will report on ancient and recent results and techniques employed to attack the question.

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