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

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.