Rational points on cubic hypersurfaces that split off two forms

TL;DR

This paper proves that certain high-dimensional cubic hypersurfaces over the rationals, which split into multiple forms, always have rational points, extending known results to new classes of forms.

Contribution

It establishes the existence of rational points on cubic hypersurfaces that split off two forms, for dimensions at least 11, and for specific form decompositions.

Findings

Rational points exist on cubic hypersurfaces with splitting forms in high dimensions.

The result applies to hypersurfaces defined by forms splitting into two parts with specified dimensions.

The paper extends previous knowledge on rational points on cubic hypersurfaces.

Abstract

We show that if $X\subseteq \mathbb{P}^{n-1}$, defined over $\mathbb{Q}$ by a cubic form that splits off two forms, with $n\geq 11$, then $X(\mathbb{Q})$ is non-empty. The same holds for an $(m_1,m_2)$-form with $m_1\geq 4$ and $m_2\geq 5$.