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

TL;DR

This paper constructs a compactification of the space of branched coverings of the sphere, linking it to rational functions and existing Hurwitz space compactifications, providing a topological and algebraic understanding of degenerations.

Contribution

It introduces a new compactification of the space of branched coverings of a surface, connecting degenerations to rational functions and existing Hurwitz space theories.

Findings

The topology on the compactified space matches the rational function coefficient topology.

The compactification coincides with the Diaz-Edidin-Natanzon-Turaev space for simple critical values.

Degenerations are characterized as locally wedge-like singular surfaces.

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\rightarrow 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 case $ \Sigma=S^2$, the topology that we define on the obtained completion $\bar{X}_{\Sigma,n}$ coincides on $X_{S^2,n}$ with the topology induced by the space of coefficients of rational functions $ P/Q $, where $ P,Q $ are homogeneous polynomials of degree $ n $ on $ \mathbb{C}\mathrm{P}^1\cong S^2$. We prove that $\bar{X}_{\Sigma,n}$ 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.