Moduli spaces for covers of the Riemann sphere (Q1320041)

The present paper simplifies and extends the construction of moduli spaces for covers of the Riemann sphere which has been constructed in joint work of the author and \textit{M. Fried} [Math. Ann. 290, 771-800 (1991; Zbl 0763.12004)]. For some Galois realizations, it is necessary to replace the Hurwitz braid group by the Artin braid group. The author studies the topological covering of the complement of the discriminant locus in affine \(r\)-space associated to the braiding action of the Artin braid group, namely the fundamental group of this complement, on certain generating systems of a finite group \(G\). He shows that this covering has a structure of \(\mathbb{Q}\)-variety and that \(\Aut(G)\) acts on this variety in a natural way. Moreover certain subspaces associated to \(r\)-tuples of conjugacy classes of \(G\) and its field of definition are considered.