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 $\Sigma \to S^2$ 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.

30 pages, in Russian, 7 figures, keywords: branched coverings, compactification of the Hurwitz space, cell structure, trigonal curves. A new example added. Final version accepted for publication in St. Petersburg Mathematical Journal

References in corpus (2)

• On deformation types of real elliptic surfaces
• Compactification of the space of branched coverings of the two-dimensional sphere