Cubic fourfolds with an involution

Abstract

There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with a symplectic involution has no associated K 3 K3 surface and is conjecturely irrational. In contrast, a cubic fourfold with a particular anti-symplectic involution has an associated K 3 K3 , and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hassett maximal. We study the algebraic and transcendental lattices for cubics with an involution both lattice theoretically and geometrically.