Hilbert squares of K3 surfaces and Debarre-Voisin varieties

Olivier Debarre; Frédéric Han; Kieran O’Grady; Claire Voisin

doi:10.5802/jep.125

Hilbert squares of K3 surfaces and Debarre-Voisin varieties
[Schémas de Hilbert ponctuels de surfaces K3 et variétés de Debarre-Voisin]

Olivier Debarre ¹ ; Frédéric Han ¹ ; Kieran O’Grady ² ; Claire Voisin ³

¹ Université de Paris, Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche 75013 Paris, France
² Sapienza Università di Roma, Dip.to di Matematica P.le A. Moro 5, 00185 Italia
³ Collège de France 3 rue d’Ulm, 75005 Paris, France

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 653-710.

Résumé
Abstract

Les variétés hyper-kählériennes de Debarre-Voisin sont construites à l’aide de $3$ -formes alternées sur un espace vectoriel complexe de dimension $10$ , que nous appelons des trivecteurs. Elles présentent de nombreuses analogies avec les variétés de Beauville-Donagi qui sont construites en partant d’une cubique de dimension $4$ . Nous étudions dans cet article différents trivecteurs dont la variété de Debarre-Voisin associée est dégénérée au sens où elle est soit réductible, soit de dimension excessive. Nous montrons que, sous une spécialisation d’un trivecteur général en de tels trivecteurs, les variétés de Debarre-Voisin correspondantes se spécialisent en des variétés hyper-kählériennes lisses, birationnellement isomorphes au schéma de Hilbert des paires de points sur une surface K3.

Debarre-Voisin hyperkähler fourfolds are built from alternating $3$ -forms on a $10$ -dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general $1$ -parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3 surface.

Reçu le : 2019-01-15
Accepté le : 2020-01-28
Publié le : 2020-04-02

DOI : 10.5802/jep.125

Classification : 14J32, 14J35, 14M15, 14J70, 14J28
Keywords: Hyperkähler fourfolds, trivectors, moduli spaces, Hilbert schemes of 2 points of a K3 surfaces
Mot clés : Variétés hyper-kählériennes, trivecteurs, espaces de modules, schémas de Hilbert ponctuels de surfaces K3

Affiliations des auteurs :

Olivier Debarre ¹ ; Frédéric Han ¹ ; Kieran O’Grady ² ; Claire Voisin ³

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Olivier Debarre and Fr\'ed\'eric Han and Kieran O{\textquoteright}Grady and Claire Voisin},
     title = {Hilbert squares of {K3} surfaces and {Debarre-Voisin} varieties},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {653--710},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.125},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.125/}
}

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Olivier Debarre; Frédéric Han; Kieran O’Grady; Claire Voisin. Hilbert squares of K3 surfaces and Debarre-Voisin varieties. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 653-710. doi : 10.5802/jep.125. https://jep.centre-mersenne.org/articles/10.5802/jep.125/

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