Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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C. Petronio - F. Sarti

Counting surface branched covers

created by sarti on 01 Aug 2019
modified on 29 Jun 2021


Published Paper

Inserted: 1 aug 2019
Last Updated: 29 jun 2021

Journal: Studia Scientiarum Mathematica Hungarica
Volume: 56
Number: 3
Pages: 14
Year: 2019

ArXiv: 1901.08316 PDF


To a branched cover f between orientable surfaces one can associate a certain branch datum D(f), that encodes the combinatorics of the cover. This D(f) satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum D there exists a branched cover f such that D(f)=D. One can actually refine this problem and ask how many these f's exist, but one must of course decide what restrictions one puts on such f's, and choose an equivalence relation up to which one regards them. And it turns out that quite a few natural choices are possible. In this short note we carefully analyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different we employ Grothendieck's dessins d'enfant.

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