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

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.