no. 4
On quotients of ℳ ¯ g,n by certain subgroups of S n
[Quotients de ℳ ¯ g,n par certaines sous-groupes de S n ]
Schwarz, Irene1
1 Humboldt Universität Berlin Institut für Mathematik Rudower Chausee 25 12489 Berlin (Germany)
Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1417-1435.

Nous analysons quand certains quotients de l’espace compactifié des modules ℳ ¯ G :=ℳ ¯ 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 ℳ ¯ G passe du type général au cas uniruled quand n augmente. Comme application, on considère la variété de différences universelles ℳ ¯ g,2n /S n ×S n .

We investigate when certain quotients of the compactified moduli space of n-pointed genus g curves ℳ ¯ G :=ℳ ¯ 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 ℳ ¯ 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 ℳ ¯ g,2n /S n ×S n .

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DOI : 10.5802/aif.3496
Classification : 14H10, 14H51
Keywords: Moduli spaces, algebraic curves, Kodaira dimension
Mot clés : Espaces de modules, courbes algébriques, dimension de Kodaira
Schwarz, Irene 1

1 Humboldt Universität Berlin Institut für Mathematik Rudower Chausee 25 12489 Berlin (Germany)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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, Tome 72 (2022) no. 4, pp. 1417-1435. doi : 10.5802/aif.3496. https://aif.centre-mersenne.org/articles/10.5802/aif.3496/

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