Moduli of G-covers of curves: geometry and singularities
Annales de l'Institut Fourier, Online first, 50 p.

We analyze the singular locus and the locus of non-canonical singularities of the moduli space ¯ g,G of curves with a G-cover for any finite group G. We show that non-canonical singularities are of two types: T-curves, that is singularities lifted from the moduli space 𝕄 ¯ g of stable curves, and J-curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case G=S 3 , the J-locus is empty, which is the first fundamental step in evaluating the Kodaira dimension of  ¯ g,S 3 .

Nous analysons le lieu singulier et le lieu des singularités non-canoniques de l’espace de modules ¯ g,G des courbes avec un G-recouvrement où G est un groupe fini. Nous montrons que les singularités non canoniques sont de deux types : T-courbes, c’est-à-dire des singularités relevées de l’espace de modules ¯ g des courbes stables, et J-courbes, c’est-à-dire des singularités nouvelles caractérisées entièrement par le graphe dual du recouvrement. Enfin, nous prouvons que dans le cas G=S 3 , le lieu J est vide, une première étage très importante dans l’évaluation de la dimension de Kodaira de ¯ g,S 3

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3503
Classification: 14H10,  14H20,  14H60,  14D20,  14D23
Keywords: Moduli of curves, G-covers, curves, stable curves, curve coverings, singularities, birational geometry.
Galeotti, Mattia 1

1 Università di Bologna Piazza di Porta S. Donato, 5 40126 Bologna (Italy)
@unpublished{AIF_0__0_0_A85_0,
     author = {Galeotti, Mattia},
     title = {Moduli of $G$-covers of curves: geometry and singularities},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2022},
     doi = {10.5802/aif.3503},
     language = {en},
     note = {Online first},
}
TY  - UNPB
TI  - Moduli of $G$-covers of curves: geometry and singularities
JO  - Annales de l'Institut Fourier
PY  - 2022
DA  - 2022///
PB  - Association des Annales de l’institut Fourier
N1  - Online first
UR  - https://doi.org/10.5802/aif.3503
DO  - 10.5802/aif.3503
LA  - en
ID  - AIF_0__0_0_A85_0
ER  - 
%0 Unpublished Work
%T Moduli of $G$-covers of curves: geometry and singularities
%J Annales de l'Institut Fourier
%D 2022
%I Association des Annales de l’institut Fourier
%Z Online first
%U https://doi.org/10.5802/aif.3503
%R 10.5802/aif.3503
%G en
%F AIF_0__0_0_A85_0
Galeotti, Mattia. Moduli of $G$-covers of curves: geometry and singularities. Annales de l'Institut Fourier, Online first, 50 p.

[1] Abramovich, Dan; Corti, Alessio; Vistoli, Angelo 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] Abramovich, Dan; Vistoli, Angelo Compactifying the space of stable maps, J. Amer. Math. Soc., Volume 15 (2002) no. 1, pp. 27-75 | DOI | MR | Zbl

[3] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Pillip A. 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] Bertin, José; Romagny, Matthieu Champs de Hurwitz, Mém. Soc. Math. Fr. (N.S.) (2011) no. 125-126, p. 219 | DOI | MR | Zbl

[5] Calmès, Baptiste; Fasel, Jean Groupes classiques, Autours des schémas en groupes. Vol. II (Panor. Synthèses), Volume 46, Soc. Math. France, Paris, 2015, pp. 1-133 | MR | Zbl

[6] Chiodo, Alessandro Stable twisted curves and their r-spin structures, Ann. Inst. Fourier, Volume 58 (2008) no. 5, pp. 1635-1689 | DOI | MR | Zbl

[7] Chiodo, Alessandro; Eisenbud, David; Farkas, Gavril; Schreyer, Frank-Olaf Syzygies of torsion bundles and the geometry of the level modular variety over ¯ g , Invent. Math., Volume 194 (2013) no. 1, pp. 73-118 | DOI | MR | Zbl

[8] Chiodo, Alessandro; Farkas, Gavril 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] Deligne, Pierre; Mumford, David 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] Eisenbud, David; Harris, Joe The Kodaira dimension of the moduli space of curves of genus 23, Invent. Math., Volume 90 (1987) no. 2, pp. 359-387 | DOI | MR | Zbl

[11] Farkas, Gavril; Jensen, David; Payne, Sam The Kodaira dimensions of ¯ 22 and ¯ 23 (2020) (https://arxiv.org/abs/2005.00622)

[12] Farkas, Gavril; Verra, Alessandro 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] Galeotti, Mattia Singularities of Moduli of Curves with a Universal Root, Doc. Math., Volume 22 (2017), pp. 1337-1373 | DOI | MR | Zbl

[14] Giraud, Jean Cohomologie non abélienne, Grundlehren der Mathematischen Wissenschaften, 179, Springer-Verlag, Berlin-New York, 1971, ix+467 pages | MR | Zbl

[15] Harris, J. 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] Harris, Joe; Mumford, David 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] Jarvis, Tyler J.; Kaufmann, Ralph; Kimura, Takashi Pointed admissible G-covers and G-equivariant cohomological field theories, Compos. Math., Volume 141 (2005) no. 4, pp. 926-978 | DOI | MR | Zbl

[18] Ludwig, Katharina On the geometry of the moduli space of spin curves, J. Algebraic Geom., Volume 19 (2010) no. 1, pp. 133-171 | DOI | Zbl

[19] Prill, David Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J., Volume 34 (1967), pp. 375-386 | DOI | Zbl

[20] Reid, Miles Canonical 3-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] Schmitt, Johannes; van Zelm, Jason Intersections of loci of admissible covers with tautological classes (2018) (https://arxiv.org/abs/1808.05817)

[22] Sernesi, Edoardo Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, 334, Springer-Verlag, Berlin, 2006, xii+339 pages | DOI | MR | Zbl

Cited by Sources: