Cubic Fourfolds with an Involution

TL;DR

This paper investigates cubic fourfolds with involutions, revealing their diverse behaviors related to rationality, including examples that are conjecturally irrational and others that are rational, based on the type of involution.

Contribution

It classifies involutions on cubic fourfolds and links their properties to rationality conjectures, introducing the concept of Hassett maximal cubics.

Findings

Symplectic involutions lead to conjecturally irrational cubics without associated K3 surfaces.

Certain anti-symplectic involutions produce rational cubics with associated K3 surfaces.

Hassett maximal cubics are contained in all non-empty Hassett divisors.

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, we show 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.