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Irreducibility of moduli spaces of cyclic unramified covers of genus \(g\) curves. (English) Zbl 0601.14022

The authors define a simple by cyclic sequence of type \((m,r,n)\) by a covering of Riemann surfaces: \(X'\to^{\psi}X\to^{\phi}{\mathbb P}^ 1\), where \(\phi\) is a simple covering of degree \(m\) with \(r\) ramification points and \(\psi\) is an étale covering of degree \(n\). They compute the Nielsen type of this covering, and they show that the space of equivalence classes of simple by cyclic sequences of type \((m,r,n)\), with \(r\geq 2m\) and \(m\geq 3\) is irreducible.
In the proof, they use the methods developed by E. Fadell and J. van Buskirk [Duke Math. J. 29, 243–257 (1962; Zbl 0122.17804)], M. Fried [Commun. Algera 5, 17–82 (1977; Zbl 0478.12006)], M. Fried and R. Biggers [J. Reine Angew. Math. 335, 87–121 (1982; Zbl 0484.14002)], and others.
Reviewer: T. Sekiguchi

MSC:

14H10 Families, moduli of curves (algebraic)
14H30 Coverings of curves, fundamental group
14D20 Algebraic moduli problems, moduli of vector bundles
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