Rational points on cubic hypersurfaces that split off a form

TL;DR

This paper proves that high-dimensional cubic hypersurfaces over rationals contain rational points if their defining form splits into two variable-disjoint forms, advancing understanding of rational solutions on such varieties.

Contribution

It establishes the existence of rational points on certain split cubic hypersurfaces of high dimension, extending previous results in the field.

Findings

Rational points exist on cubic hypersurfaces of dimension 11 or more that split into two forms

The form splitting condition is sufficient for rational points to be present

Includes an appendix on the unramified Brauer group of singular cubic hypersurfaces

Abstract

Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms that share no common variables. ----- This paper features an appendix "Groupe de Brauer non ramifi\'e des hypersurfaces cubiques singuli\`eres (d'apr\`es P. Salberger)", by J.-L. Colliot-Th\'l\`ene.