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Tr. Mat. Inst. Steklova, 2017, Volume 298, Pages 127–138 (Mi tm3815)  

Compactification of the Space of Branched Coverings of the Two-Dimensional Sphere

V. I. Zvonilova, S. Yu. Orevkovbcd

a Chukotka Branch of the North-Eastern Federal University, Studencheskaya ul. 3, Anadyr, Chukotka, 689000 Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
c Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France
d National Research University "Higher School of Economics," ul. Myasnitskaya 20, Moscow, 101000 Russia

Abstract: For a closed oriented surface $\Sigma $ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let $X_{\Sigma ,n}$ be the set of isomorphism classes of orientation-preserving $n$-fold branched coverings $\Sigma \to S^2$ of the two-dimensional sphere. We complete $X_{\Sigma ,n}$ with the isomorphism classes of mappings that cover the sphere by the degenerations of $\Sigma $. In the case $\Sigma =S^2$, 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 $\mathbb C\mathrm P^1\cong S^2$. We prove that $\overline {X}_{\Sigma ,n}$ 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.

Funding Agency Grant Number
Russian Science Foundation 14-21-00053
The work of the second author is supported by the Russian Science Foundation under grant 14-21-00053.


DOI: https://doi.org/10.1134/S0371968517030098

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English version:
Proceedings of the Steklov Institute of Mathematics, 2017, 298, 118–128

Bibliographic databases:

UDC: 515.179.25
Received: November 1, 2016

Citation: V. I. Zvonilov, S. Yu. Orevkov, “Compactification of the Space of Branched Coverings of the Two-Dimensional Sphere”, Complex analysis and its applications, Collected papers. On the occasion of the centenary of the birth of Boris Vladimirovich Shabat, 85th anniversary of the birth of Anatoliy Georgievich Vitushkin, and 85th anniversary of the birth of Andrei Aleksandrovich Gonchar, Tr. Mat. Inst. Steklova, 298, MAIK Nauka/Interperiodica, Moscow, 2017, 127–138; Proc. Steklov Inst. Math., 298 (2017), 118–128

Citation in format AMSBIB
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\paper Compactification of the Space of Branched Coverings of the Two-Dimensional Sphere
\inbook Complex analysis and its applications
\bookinfo Collected papers. On the occasion of the centenary of the birth of Boris Vladimirovich Shabat, 85th anniversary of the birth of Anatoliy Georgievich Vitushkin, and 85th anniversary of the birth of Andrei Aleksandrovich Gonchar
\serial Tr. Mat. Inst. Steklova
\yr 2017
\vol 298
\pages 127--138
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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