Abstract
For a closed oriented surface Σ \Sigma , let X Σ , n X_{\Sigma , n} be the space of isomorphism classes of n n -fold orientation preserving branched coverings Σ → S 2 \Sigma \rightarrow S ^2 of the two-dimensional sphere. Earlier, the authors constructed a compactification X ¯ Σ , n \overline {X}_{\Sigma , n} of this space, which coincides with the Diaz–Edidin–Natanzon–Turaev compactification of the Hurwitz space H ( Σ , n ) ⊂ X Σ , n H (\Sigma , n) \subset X_{\Sigma , n} that consists of isomorphism classes of branched coverings with all critical values being simple. With the help of Grothendieck’s dessins d’enfants, a cellular structure of this compactification is constructed. The results obtained are applied to the space of trigonal curves on an arbitrary Hirzebruch surface.
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