Involutions and linear systems on holomorphic symplectic manifolds

Abstract.

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 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 $\mathbb{P}^{5} .$ 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 antisymplectic involutions generating an interesting dynamical system, a case Strange duality and what is probably an involution on the moduli space degree-2 quasi-polarized (X, H) where X is a deformation of (K3)^[2].