Special Cubic Fourfolds

Abstract

A cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfolds form a countably infinite union of irreducible families ${\mathcal F}$$_d$, each a divisor in the moduli space ${\mathcal F}$ of cubic fourfolds. For an infinite number of these families, the Hodge structure on the nonspecial cohomology of the cubic fourfold is essentially the Hodge structure on the primitive cohomology of a K3 surface. We say that this K3 surface is associated to the special cubic fourfold. In these cases, ${\mathcal F}$$_d$ is related to the moduli space ${\mathcal N}$$_d$ of degree d K3 surfaces. In particular, ${\mathcal C}$ contains infinitely many moduli spaces of polarized K3 surfaces as closed subvarieties. We can often construct a correspondence of rational curves on the special cubic fourfold parametrized by the K3 surface which induces the isomorphism of Hodge structures. For infinitely many values of d, the Fano variety of lines on the generic cubic fourfold of ${\mathcal C}$$_d$ is isomorphic to the Hilbert scheme of length-two subschemes of an associated K3 surface.