Cubic fourfolds of discriminant 24 and rationality

TL;DR

This paper investigates cubic fourfolds with discriminant 24, linking their algebraic cycles and Brauer group elements to their rationality, and providing new examples of rational fourfolds.

Contribution

It establishes a criterion for rationality based on the vanishing of the Brauer class and constructs new rational examples of cubic fourfolds with discriminant 24.

Findings

Rationality of cubic fourfolds correlates with vanishing Brauer class.

Constructs countably-infinite new rational cubic fourfolds.

Identifies special algebraic cycles related to discriminant 24.

Abstract

Cubic fourfolds of discriminant 24 contain special codimension-two algebraic cycles of degree 6 and self-intersection 20. Such cycles may be represented by singular scrolls or del Pezzo surfaces. A discriminant 24 cubic fourfold gives rise to a twisted surface, consisting of a degree-six K3 surface and a two-torsion element of its Brauer group. We show that the cubic fourfold is rational if the Brauer class vanishes. This yields a countably-infinite collection of new examples of rational cubic fourfolds, each of codimension two in moduli.