Hilbert squares of K3 surfaces and Debarre-Voisin varieties

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

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

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.

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

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