On quotients of ¯ g,n by certain subgroups of S n
Annales de l'Institut Fourier, Online first, 19 p.

We investigate when certain quotients of the compactified moduli space of n-pointed genus g curves ¯ G := ¯ g,n /G are of general type or, on the contrary, uniruled, for a fairly broad class of subgroups G of the symmetric group S n which act by permuting the n marked points. We show that the property of being of general type only depends on the transpositions contained in G. Furthermore, in the case that G is the full symmetric group S n or a product S n 1 ××S n m , we find a narrow transitional band in which ¯ G changes its behaviour from being of general type to its opposite, i.e. being uniruled, as n increases. As an application we consider the universal difference variety ¯ g,2n /S n ×S n .

Nous analysons quand certains quotients de l’espace compactifié des modules ¯ G := ¯ g,n /G de courbes de genre g marquées en n points sont de type général, ou au contraire, uniruled, pour une classe assez large de sous-groupes G du groupe symétrique S n agissant par permutation des points marqués. On montre que la propriété d’être de type général ne dépend que des transpositions contenues dans G. Dans le cas où G est le groupe symétrique S n ou un produit S n 1 ××S n m on trouve une bande étroite de transition où ¯ G passe du type général au cas uniruled quand n augmente. Comme application, on considère la variété de différences universelles ¯ g,2n /S n ×S n .

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Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3496
Classification: 14H10,  14H51
Keywords: Moduli spaces, algebraic curves, Kodaira dimension
Schwarz, Irene 1

1 Humboldt Universität Berlin Institut für Mathematik Rudower Chausee 25 12489 Berlin (Germany)
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Schwarz, Irene. On quotients of $\protect \,\hspace{1.111pt}\protect \overline{\protect \!\hspace{-1.111pt}\protect \mathcal{M}}_{g,n}$ by certain subgroups of $S_n$. Annales de l'Institut Fourier, Online first, 19 p.

[1] Arbarello, Enrico; Cornalba, Maurizio The Picard groups of the moduli spaces of curves, Topology, Volume 26 (1987) no. 2, pp. 153-171

[2] Arbarello, Enrico; Cornalba, Maurizio Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Publ. Math., Inst. Hautes Étud. Sci., Volume 88 (1998) no. 1, pp. 97-127

[3] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip Geometry of algebraic curves: Volume II. With a contribution by Joseph Daniel Harris, Grundlehren der Mathematischen Wissenschaften, 268, Springer, 2011

[4] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip; Harris, Joe Geometry of algebraic curves: Volume I, Grundlehren der Mathematischen Wissenschaften, 267, Springer, 1985

[5] Casnati, Gianfranco; Fontanari, Claudio On the rationality of moduli spaces of pointed curves, J. Lond. Math. Soc., Volume 75 (2007) no. 3, pp. 582-596

[6] Eisenbud, David; Harris, Joe The Kodaira dimension of the moduli space of curves of genus23, Invent. Math., Volume 90 (1987) no. 2, pp. 359-387

[7] Farkas, Gavril Koszul divisors on moduli spaces of curves, Am. J. Math., Volume 131 (2009) no. 3, pp. 819-867

[8] Farkas, Gavril; Jensen, David; Payne, Sam The Kodaira dimensions of M22 and M23 (2020) (https://arxiv.org/abs/2005.00622)

[9] Farkas, Gavril; Verra, Alessandro The classification of universal Jacobians over the moduli space of curves, Comment. Math. Helv., Volume 88 (2013) no. 3, pp. 587-611

[10] Farkas, Gavril; Verra, Alessandro The universal difference variety over ¯ g , Rend. Circ. Mat. Palermo, Volume 62 (2013) no. 1, pp. 97-110

[11] Farkas, Gavril; Verra, Alessandro The universal theta divisor over the moduli space of curves, J. Math. Pures Appl., Volume 100 (2013) no. 4, pp. 591-605

[12] Farkas, Gavril; Verra, Alessandro The geometry of the moduli space of odd spin curves, Ann. Math., Volume 180 (2014) no. 3, pp. 927-970

[13] Harris, Joe; Mumford, David On the Kodaira dimension of the moduli space of curves, Invent. Math., Volume 67 (1982) no. 1, pp. 23-86

[14] Kouvidakis, Alexis On some results of Morita and their application to questions of ampleness, Math. Z., Volume 241 (2002) no. 1, pp. 17-33

[15] Logan, Adam The Kodaira dimension of moduli spaces of curves with marked points, Am. J. Math., Volume 125 (2003) no. 1, pp. 105-138

[16] Massarenti, Alex The automorphism group of ¯ g,n , J. Lond. Math. Soc., Volume 89 (2014) no. 1, pp. 131-150

[17] Schwarz, Irene On the Kodaira dimension of the moduli space of nodal curves (2018) (https://arxiv.org/abs/1811.01193)

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