Involutions and linear systems on holomorphic symplectic manifolds

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 $(K3)^{[2]}$.Comment: 42 page

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

Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces 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 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 an irreducible symplectic 4-fold numerically equivalent to the Hilbert square of a K3-surface, that H is an ample divisor on X of square 2 for Beauville's quadratic form and that the map associated to |H| is the composition of the quotient map $X\to Y$ for an anti-symplectic involution on X followed by an immersion of Y; then Y is an EPW-sextic and $X\to Y$ is the natural double cover.Comment: 29 page

Decomposable cycles and Noether-Lefschetz loci

We prove that there exist smooth surfaces of degree d in projective 3-space such that the group of rational equivalence classes of decomposable 0-cycles has rank at least the integer part of (d-1)/3.Comment: Exposition improve

Computations with modified diagonals

Beauville and Voisin proved that the third modified diagonal of a complex K3 surface X represents a torsion class in the Chow group of X^3. Motivated by this result and by conjectures of Beauville and Voisin on the Chow ring of hyperkaehler varieties we prove some results on modified diagonals of projective varieties and we formulate a conjecture.Comment: Minor correction

Double covers of EPW-sextics

EPW-sextics are special 4-dimensional sextic hypersurfaces (with 20 moduli) which come equipped with a double cover. We analyze the double cover of EPW-sextics parametrized by a certain prime divisor in the moduli space. We associate to the generic sextic parametrized by that divisor a K3 surface of genus 6 and we show that the double epw sextic is a contraction of the Hilbert square of the K3. This result has two consequences. First it gives a new proof of the following result of ours: smooth double EPW-sextics form a locally complete family of hyperkaehler projective deformations of the Hilbert square of a K3. Secondly it shows that away from another prime divisor in the moduli space the period map for double EPW-sextics is as well-behaved as it possibly could be.Comment: Exposition improved according to referee's request