- Published 2005

Eisenbud Popescu and Walter have constructed certain special sextic hypersurfaces in P as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3 surface (K3) 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) of the Fano variety of lines on a cubic 4-fold. Conversely suppose that X is a numerical (K3), that H is an ample divisor on X of square 2 for Beauville’s quadratic form and that the map X 99K |H | is the composition of the quotient X → Y for an anti-symplectic involution on X followed by an immersion Y →֒ |H |∨; then Y is an EPW-sextic and X → Y is the natural double cover. If a conjecture on the behaviour of certain linear systems holds this result together with previous results of ours implies that every numerical (K3) is a deformation of (K3).

@inproceedings{OGrady2005IrreducibleS4,
title={Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics},
author={Kieran G. O’Grady},
year={2005}
}