  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 $\PP^5$. 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 $(K3)^{[2]}$.
