Degenerating families of finite branched coverings (Q1866574)

For a connected complex manifold \(M\), a finite branched covering of \(M\) is a finite proper holomorphic mapping \(f:X\to M\) of an irreducible normal complex space \(X\) onto \(M\). The first named author of this paper studied the moduli space of equivalence classes of finite branched coverings of the projective line \(P^1\) in his previous paper [\textit{M. Namba}, `Families of meromorphic functions on compact Riemann surfaces', Springer, Berlin (1979; Zbl 0417.32008)]. To compactify the moduli space of equivalence classes of finite branched coverings, we have to consider degeneration of branched coverings. The authors study degenerate families of finite branched coverings of \(P^1\) and the \(m\)-dimensional complex projective space \(P^m\). The authors first introduce a picture which topologically represents a finite branched covering of the complex projective line. The authors call such a picture a Klein picture. The authors prove, among other things, that the topological structure of a degenerating family of finite brached coverings of \(P^1\) can be determined by the permutation monodromy of the general fiber and the braid monodromy of the family.