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

Abstract

Eisenbud, Popescu, and Walter [4] have constructed certain special sextic hypersurfaces in ${\mathbb P}^5$ 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 $X\dashrightarrow|H|^{\vee}$ is the composition of the quotient $X\to Y$ for an antisymplectic involution on $X$ followed by an immersion $Y\hookrightarrow|H|^{\vee}$; then $Y$ is an EPW sextic, and $X\to Y$ is the natural double cover