## 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)

### Keywords:

moduli spaces; Hurwitz spaces; algebraic stacks

### Citations:

Zbl 0181.48803; Zbl 0814.14030; Zbl 0860.14024
Full Text:

### References:

 [1] Abramovich D., Proc. Amer. Math. Soc. 131 pp 685– (2003) · Zbl 1037.14008 [2] Abramovich D., Progress in Math. 181, in: Resolution of Singularities. A research textbook in tribute to Oscar Zariski (2000) [3] Abramovich D., Contemp. Math. 276, in: Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000) pp 89– (2001) [4] DOI: 10.1090/S0894-0347-01-00380-0 · Zbl 0991.14007 [5] Boggi M., Compositio Math. 120 pp 171– (2000) · Zbl 0959.14010 [6] Chen W., Contemporary Math. 310, in: Orbifolds in Mathematics and Physics (2002) [7] DOI: 10.1007/BF02684599 · Zbl 0181.48803 [8] Ekedahl T., Progr. Math. 129, in: The Moduli Space of Curves (Texel Island, 1994) pp 173– (1995) [9] Fulton, W. and Pandharipande, R. Notes on stable maps and quantum cohomology. Proc. Sympos. Pure Math., Part 2. 1995, Santa Cruz. Algebraic Geometry, pp.45–96. Providence, RI: Amer. Math. Soc. · Zbl 0898.14018 [10] Grothendieck A., Inst. Hautes Études Sci. Publ. Math. 8 pp 222– (1961) [11] Grothendieck, A. 1962. Fondements de la géométrie algébrique. Extraits du Sém. Bourbaki. 1962, Paris. 1957–1962. Secrétariat Math [12] Harris J., Invent. Math. 1982 pp 23– (1982) · Zbl 0506.14016 [13] Illusie L., Lecture Notes in Mathematics 239 & Lecture Notes in Mathematics 283, in: Complexe Cotangent et Déformations. I, II (1971) [14] Kato K., Kodai Math. J. 22 pp 161– (1999) · Zbl 0957.14015 [15] Kontsevich M., Progr. Math. 129, in: The Moduli Space of Curves (Texel Island, 1994) pp 335– (1995) [16] Laumon G., Ergebnisse der Mathematik und ihrer Grenzgebiete 39, in: Champs Algébriques (2000) [17] Looijenga E., J. Algebraic Geom. 3 pp 283– (1994) [18] Milne J. S., Étale Cohomology (1980) [19] Mochizuki S., Publ. Res. Inst. Math. Sci. 31 pp 355– (1995) · Zbl 0866.14013 [20] Mumford D., Progr. Math. 36, in: Arithmetic and Geometry pp 271– (1983) [21] Pikaart M., Progr. Math. 129, in: The moduli space of curves (Texel Island, 1994) pp 483– (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.