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 symplectic involution has no associated K3 surface and is conjecturely irrational. In contrast, a cubic fourfold with a particular anti-symplectic involution has an associated 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.

Involutions on K3 surfaces and derived equivalence

“…• Higher-dimensional analogs of Enriques involutions are studied in [OS11]. • Involutions on cubic fourfolds -both symplectic (see [LZ22] and [HT10]) and anti-symplectic -are studied in [Mar22]. The corresponding actions on lattices are described explicitly.…”