Compactification of the space of branched coverings of the two-dimensional sphere


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For a closed oriented surface Σ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let XΣ,n be the set of isomorphism classes of orientation-preserving n-fold branched coverings Σ → S2 of the two-dimensional sphere. We complete XΣ,n with the isomorphism classes of mappings that cover the sphere by the degenerations of Σ. In the case Σ = S2, the topology that we define on the obtained completion \({\overline X _{\Sigma ,n}}\) coincides on \({X_{{s^2},n}}\) with the topology induced by the space of coefficients of rational functions P/Q, where P and Q are homogeneous polynomials of degree n on ℂP1S2. We prove that \({\overline X _{\Sigma ,n}}\) coincides with the Diaz–Edidin–Natanzon–Turaev compactification of the Hurwitz space H(Σ, n) ⊂ XΣ,n consisting of isomorphism classes of branched coverings with all critical values being simple.

Sobre autores

V. Zvonilov

Chukotka Branch of the North-Eastern Federal University

Autor responsável pela correspondência
Email: zvonilov@gmail.com
Rússia, Studencheskaya ul. 3, Anadyr, Chukotka, 689000

S. Orevkov

Steklov Mathematical Institute of Russian Academy of Sciences; Institut de Mathématiques de Toulouse; National Research University “Higher School of Economics,”

Email: zvonilov@gmail.com
Rússia, ul. Gubkina 8, Moscow, 119991; 118 route de Narbonne, Toulouse Cedex 9, F-31062; ul. Myasnitskaya 20, Moscow, 101000

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