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

Abstract

For a closed oriented surface Σ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let X_Σ,n be the set of isomorphism classes of orientation-preserving n-fold branched coverings Σ → S² of the two-dimensional sphere. We complete X_Σ,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 ℂP¹ ≌ S². We prove that \({\overline X _{\Sigma ,n}}\) coincides with the Diaz–Edidin–Natanzon–Turaev compactification of the Hurwitz space H(Σ, n) ⊂ X_Σ,n consisting of isomorphism classes of branched coverings with all critical values being simple.