A topological construction of families of Galois covers of the line (Q6593005)

Consider a finite group \(G\) acting on smooth projective curves \(C,C'\) over the complex numbers. Say that \(C\) and \(C'\) are \textit{topologically equivalent} if there exists an orientation-preserving homeomorphism \(f:C\to C'\) which is \(\alpha\)-equivariant for some \(\alpha\in \Aut(G)\). The \(G\)-curves \(C\) and \(C'\) are \textit{isomorphic} if one can find such a map \(f\) which is a biholomorphism.\N\NGiven a \(G\)-covering \(C\to \mathbb P^1\), the authors give a new construction of an algebraic family of \(G\)-curves such that\N\begin{itemize}\N\item[1.] every \(G\)-curve \(C'\) in the family is topologically equivalent to \(C\) ;\N\item[2.] every \(G\)-curve topologically equivalent to \(C\) is \(G\)-isomorphic to some fiber of the family and to at most a finite number of fibers.\N\end{itemize}\N\NThis is an alternative to a previous construction due to González-Díez and Harvey and it corrects an inaccuracy therein when \(G\) is non-abelian. In the opposite case where the Galois group has trivial center, the authors recover some results due to Fried and Völklein.\N\NThe construction is based on basic properties of configuration spaces, covering theory, and the Grauert-Remmert extension theorem.