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Eisenbud Popescu and Walter have constructed certain special sextic hypersurfaces in P 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 |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)[2] is a deformation of (K3) [2] .

A K3 surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple (X, H) with X a deformation of (K3)([n]) and H an ample divisor of square 2 for Beauville's quadratic form. If n = 2 then according to the conjecture X is a double cover of a (singular) sextic 4-fold in P-5. It follows from the conjecture that a deformation of (K3)([n]) carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture. In doing so we bump into some interesting geometry: examples of two anti-symplectic involutions generating an interesting dynamical system, a case of Strange duality and what is probably an involution on the moduli space of degree-2 quasi-polarized (X, H) where X is a deformation of (K3)([2])

(0.0.9) (We will denote Σ by Σ(V ) whenever we will need to keep track of V , and similarly for ∆). Then Σ and ∆ are closed subsets of LG( 3 V ); a straightforward computation shows that Σ and ∆ are irreducible of codimension 1 -see Section 2 for the case of Σ. LetLG(In [15] we proved the following results. If A ∈LG( 3 V ) 0 then Y A = P(V ) and there exists a finite degree-2 map f A : X A → Y A unramified over