Cassels–Swinnerton-Dyer conjecture for cubic threefolds

Establish the threefold case of the Cassels–Swinnerton-Dyer conjecture, which asserts the relevant rationality or unirationality conclusion for cubic threefolds and remains unresolved according to the paper.

Background

The paper uses the recently proved Cassels–Swinnerton-Dyer conjecture for cubic surfaces as its principal arithmetic input. That result states that an irreducible cubic surface possessing an effective zero-cycle of degree coprime to three has a rational point.

The authors note that the analogous statement for cubic threefolds is still unresolved. They nevertheless obtain a complete classification of equivariant unirationality for cubic threefolds by combining the cubic-surface result with equivariant geometric methods, including Pfaffian representations and Condition (A*).