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Twisted bundles and admissible covers. (English) Zbl 1077.14034

The main result in this article is that the stack \({\mathcal B}^{bal}_{g,n}(G)\) is a smooth scheme for a suitable finite group \(G\). The resulting morphism \({\mathcal B}^{bal}_g(G) \to \overline {\mathcal M}_g\) is a Galois cover with automorphism group Aut\((G)\). This result implies that all local computations (in the étale topology) on the stack \(\overline{\mathcal M}_g\) can be carried out on the smooth scheme \({\mathcal B}^{bal}_g(G)\).
The authors review their work on twisted \(G\)-covers, show the equivalence of the stacks \({\mathcal B}^{bal}_{g,n}(G)\) and \({\mathcal A}dm_{g,n}(G)\) The article is as well a survey on the theory of twisted \(G\)-covers, provides new results, and relates the topic to the work of P. Deligne and D. Mumford [Publ. Math., Inst. Hautes Étud. Sci. 36, 75–109 (1969; Zbl 0181.48803)], E. Looijenga [J. Algebr. Geom. 3, 283–293 (1994; Zbl 0814.14030)] and M. Pikaart and A. J. de Jong [in: The moduli space of curves, Proc. Conf. Texel Island 1994, Prog. Math. 129, 483–509 (1995; Zbl 0860.14024)]. Furthermore, they provide an “Algebra-to-Analysis Translation Table” which also makes the results accessible to complex geometers.

MSC:

14H10 Families, moduli of curves (algebraic)
14H30 Coverings of curves, fundamental group
14A20 Generalizations (algebraic spaces, stacks)
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