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

Eisenbud Popescu and Walter have constructed certain special sextic hypersurfaces in $\PP^5$ 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 $(K3)^{[2]}$ 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) of the Fano variety of lines on a cubic $4$-fold. Conversely suppose that $X$ is a numerical $(K3)^{[2]}$, that $H$ is an ample divisor on $X$ of square $2$ for Beauville's quadratic form and that the map $X\dashrightarrow|H|^{\vee}$ is the composition of the quotient $X\to Y$ for an anti-symplectic involution on $X$ followed by an immersion $Y\hra|H|^{\vee}$; then $Y$ is an EPW-sextic and $X\to Y$ is the natural double cover.