We analyze the singular locus and the locus of non-canonical singularities of the moduli space of curves with a -cover for any finite group . We show that non-canonical singularities are of two types: -curves, that is singularities lifted from the moduli space of stable curves, and -curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case , the -locus is empty, which is the first fundamental step in evaluating the Kodaira dimension of .
Nous analysons le lieu singulier et le lieu des singularités non-canoniques de l’espace de modules des courbes avec un -recouvrement où est un groupe fini. Nous montrons que les singularités non canoniques sont de deux types : -courbes, c’est-à-dire des singularités relevées de l’espace de modules des courbes stables, et -courbes, c’est-à-dire des singularités nouvelles caractérisées entièrement par le graphe dual du recouvrement. Enfin, nous prouvons que dans le cas , le lieu est vide, une première étage très importante dans l’évaluation de la dimension de Kodaira de
Revised:
Accepted:
Published online:
Keywords: Moduli of curves, $G$-covers, curves, stable curves, curve coverings, singularities, birational geometry.
Mot clés : Modules de courbes, $G$-covers, courbes, courbes stables, recouvrements de courbes, singularités, géométrie birationelle.
Galeotti, Mattia 1
@article{AIF_2022__72_6_2191_0, author = {Galeotti, Mattia}, title = {Moduli of $G$-covers of curves: geometry and singularities}, journal = {Annales de l'Institut Fourier}, pages = {2191--2240}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {72}, number = {6}, year = {2022}, doi = {10.5802/aif.3503}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3503/} }
TY - JOUR AU - Galeotti, Mattia TI - Moduli of $G$-covers of curves: geometry and singularities JO - Annales de l'Institut Fourier PY - 2022 SP - 2191 EP - 2240 VL - 72 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3503/ DO - 10.5802/aif.3503 LA - en ID - AIF_2022__72_6_2191_0 ER -
%0 Journal Article %A Galeotti, Mattia %T Moduli of $G$-covers of curves: geometry and singularities %J Annales de l'Institut Fourier %D 2022 %P 2191-2240 %V 72 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3503/ %R 10.5802/aif.3503 %G en %F AIF_2022__72_6_2191_0
Galeotti, Mattia. Moduli of $G$-covers of curves: geometry and singularities. Annales de l'Institut Fourier, Volume 72 (2022) no. 6, pp. 2191-2240. doi : 10.5802/aif.3503. https://aif.centre-mersenne.org/articles/10.5802/aif.3503/
[1] Twisted bundles and admissible covers, Comm. Algebra, Volume 31 (2003) no. 8, pp. 3547-3618 (Special issue in honor of Steven L. Kleiman) | DOI | MR | Zbl
[2] Compactifying the space of stable maps, J. Amer. Math. Soc., Volume 15 (2002) no. 1, pp. 27-75 | DOI | MR | Zbl
[3] Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften, 268, Springer, Heidelberg, 2011, xxx+963 pages (With a contribution by Joseph Daniel Harris) | DOI | MR | Zbl
[4] Champs de Hurwitz, Mém. Soc. Math. Fr. (N.S.) (2011) no. 125-126, p. 219 | DOI | MR | Zbl
[5] Groupes classiques, Autours des schémas en groupes. Vol. II (Panor. Synthèses), Volume 46, Soc. Math. France, Paris, 2015, pp. 1-133 | MR | Zbl
[6] Stable twisted curves and their -spin structures, Ann. Inst. Fourier, Volume 58 (2008) no. 5, pp. 1635-1689 | DOI | MR | Zbl
[7] Syzygies of torsion bundles and the geometry of the level modular variety over , Invent. Math., Volume 194 (2013) no. 1, pp. 73-118 | DOI | MR | Zbl
[8] Singularities of the moduli space of level curves, J. Eur. Math. Soc. (JEMS), Volume 19 (2017) no. 3, pp. 603-658 | DOI | MR | Zbl
[9] The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969) no. 36, pp. 75-109 | MR | Zbl
[10] The Kodaira dimension of the moduli space of curves of genus , Invent. Math., Volume 90 (1987) no. 2, pp. 359-387 | DOI | MR | Zbl
[11] The Kodaira dimensions of and (2020) (https://arxiv.org/abs/2005.00622)
[12] The geometry of the moduli space of odd spin curves, Ann. of Math. (2), Volume 180 (2014) no. 3, pp. 927-970 | DOI | MR | Zbl
[13] Singularities of Moduli of Curves with a Universal Root, Doc. Math., Volume 22 (2017), pp. 1337-1373 | DOI | MR | Zbl
[14] Cohomologie non abélienne, Grundlehren der Mathematischen Wissenschaften, 179, Springer-Verlag, Berlin-New York, 1971, ix+467 pages | MR | Zbl
[15] On the Kodaira dimension of the moduli space of curves. II. The even-genus case, Invent. Math., Volume 75 (1984) no. 3, pp. 437-466 | DOI | MR | Zbl
[16] On the Kodaira dimension of the moduli space of curves, Invent. Math., Volume 67 (1982) no. 1, pp. 23-88 (With an appendix by William Fulton) | DOI | Zbl
[17] Pointed admissible -covers and -equivariant cohomological field theories, Compos. Math., Volume 141 (2005) no. 4, pp. 926-978 | DOI | MR | Zbl
[18] On the geometry of the moduli space of spin curves, J. Algebraic Geom., Volume 19 (2010) no. 1, pp. 133-171 | DOI | Zbl
[19] Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J., Volume 34 (1967), pp. 375-386 | DOI | Zbl
[20] Canonical -folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 273-310 | Zbl
[21] Intersections of loci of admissible covers with tautological classes (2018) (https://arxiv.org/abs/1808.05817)
[22] Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, 334, Springer-Verlag, Berlin, 2006, xii+339 pages | DOI | MR | Zbl
Cited by Sources: