no. 3
Rational points of quiver moduli spaces
[Points rationnels des variétés de carquois]
Hoskins, Victoria1 ; Schaffhauser, Florent2
1 Freie Universität Berlin Arnimallee 3, Raum 011 14195 Berlin, Germany
2 Universidad de Los Andes Carrera 1 #18A-12 111 711 Bogotá, Colombia
Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1259-1305.

Etant donné un corps parfait k et une clôture algébrique k¯ de k, les espaces de modules de k¯-représentations semistables d’un carquois Q sont des k-variétés algébriques dont nous étudions ici les propriétés arithmétiques, en particulier les points rationnels et leur interprétation modulaire. Outre les représentations à coefficients dans k, apparaissent naturellement certaines représentations rationnelles dites tordues, à coefficients dans une algèbre à division définie sur k et qui donnent lieu à différentes k-formes de la variété des modules initiale. En guise d’application, on montre qu’une k¯-représentation stable du carquois Q est définissable sur une algèbre à division centrale bien précise, elle-même définie sur le corps des modules de la représentation considérée.

Avertissement :

Suite à une erreur de la rédaction, il manque dans la version publiée de cet article des informations sur les financements du second auteur. Le second auteur est soutenu par la Convocatoria 2018-2019 de la Facultad de Ciencias (Uniandes), Programa de investigación « Geometría y Topología de los Espacios de Módulos », le programme de recherche et d’innovation Horizon 2020 de l’Union européenne (subvention n°795222), et L’Institut d’Études Avancées de l’Université de Strasbourg (USIAS).

Compléments :
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For a perfect base field k, we investigate arithmetic aspects of moduli spaces of quiver representations over k: we study actions of the absolute Galois group of k on the k¯-valued points of moduli spaces of quiver representations over k and we provide a modular interpretation of the fixed-point set using quiver representations over division algebras, which we reinterpret using moduli spaces of twisted quiver representations (we show that those spaces provide different k-forms of the initial moduli space of quiver representations). Finally, we obtain that stable k¯-representations of a quiver are definable over a certain central division algebra over their field of moduli.

Disclaimer:

Due to an editorial error, the published version of this article lacks information on the second author’s funding. The second author is supported by Convocatoria 2018-2019 de la Facultad de Ciencias (Uniandes), Programa de investigación “Geometría y Topología de los Espacios de Módulos”, the European Union’s Horizon 2020 research and innovation programme under grant agreement No 795222 and the University of Strasbourg Institute of Advanced Study (USIAS).

Supplementary Materials:
Supplementary material for this article is supplied as a separate file:

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DOI : 10.5802/aif.3334
Classification : 14D20, 14L24, 16G20
Keywords: Algebraic moduli problems, Geometric Invariant Theory, Representations of quivers
Mots-clés : Problèmes de modules en géométrie algébrique, Théorie Géométrique des Invariants, Représentations de carquois

Hoskins, Victoria 1 ; Schaffhauser, Florent 2

1 Freie Universität Berlin Arnimallee 3, Raum 011 14195 Berlin, Germany
2 Universidad de Los Andes Carrera 1 #18A-12 111 711 Bogotá, Colombia
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Hoskins, Victoria; Schaffhauser, Florent. Rational points of quiver moduli spaces. Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1259-1305. doi : 10.5802/aif.3334. https://aif.centre-mersenne.org/articles/10.5802/aif.3334/

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