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arXiv · 2002.12402

A cell structure of the space of branched coverings of the two-dimensional sphere

Orevkov S. Yu·V. I. Zvonilov
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Abstract

For a closed oriented surface $ \Sigma $ let $X_{\Sigma,n}$ be the space of isomorphism classes of orientation preserving $n$-fold branched coverings $ \Sigma\rightarrow S^2 $ of the two-dimensional sphere. At a previous paper, the authors constructed a compactification $\bar{X}_{\Sigma,n}$ of the space that coincides with the Diaz-Edidin-Natanzon-Turaev compactification of the Hurwitz space $H(\Sigma,n)\subset X_{\Sigma,n}$ consisting of isomorphism classes of branched coverings with all critical values being simple. Using Grothendieck's dessins d'enfants we construct a cell structure of the compactification. The obtained results are applied to the space of trigonal curves on a Hirzebruch surface.

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Orevkov S. Yu, V. I. Zvonilov. 2020-02-27. A cell structure of the space of branched coverings of the two-dimensional sphere. https://doi.org/10.1090/spmj/1675

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