A cell structure of the space of branched coverings of the two-dimensional sphere
For a closed oriented surface $ Σ$ let $X_{Σ,n}$ be the space of isomorphism classes of orientation preserving $n$-fold branched coverings $ Σ\rightarrow S^2 $ of the two-dimensional sphere. At a previous paper, the authors constructed a compactification $\bar{X}_{Σ,n}$ of the space that coincides with the Diaz-Edidin-Natanzon-Turaev compactification of the Hurwitz space $H(Σ,n)\subset X_{Σ,n}$ 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.
math.AG↗