Duke Mathematical Journal

Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics

Kieran G. O'Grady

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Eisenbud, Popescu, and Walter [4] have constructed certain special sextic hypersurfaces in P5 as Lagrangian degeneracy loci. We prove that the natural double cover of a generic Eisenbud-Popescu-Walter (EPW) sextic is a deformation of the Hilbert square of a K3-surface (K3)[2] and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1); thus we get an example similar to that (discovered by Beauville and Donagi [2]) of the Fano variety of lines on a cubic 4-fold. Conversely, suppose that X is a numerical (K3)[2], suppose that H is an ample divisor on X of square 2 for Beauville's quadratic form, and suppose that the map X|H| is the composition of the quotient XY for an antisymplectic involution on X followed by an immersion Y|H|; then Y is an EPW sextic, and XY is the natural double cover

Article information

Duke Math. J., Volume 134, Number 1 (2006), 99-137.

First available in Project Euclid: 4 July 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J35: $4$-folds
Secondary: 14J10: Families, moduli, classification: algebraic theory 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry


O'Grady, Kieran G. Irreducible symplectic $4$ -folds and Eisenbud-Popescu-Walter sextics. Duke Math. J. 134 (2006), no. 1, 99--137. doi:10.1215/S0012-7094-06-13413-0. https://projecteuclid.org/euclid.dmj/1152018505

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