Rational lines on cubic hypersurfaces II

TL;DR

This paper improves bounds on the dimension of rational cubic hypersurfaces over number fields that contain rational lines, utilizing advanced number theory and algebraic geometry techniques.

Contribution

It reduces the known lower bounds for the dimension of rational cubic hypersurfaces that contain a rational line, especially over the rationals, by applying new results on quadratic and cubic forms.

Findings

For dimension ≥33 over any number field, rational cubic hypersurfaces contain a rational line.

Over the rationals, the bound is reduced to dimension ≥29.

Utilizes results on quadratic forms over quadratic extensions and recent work on cubic forms over imaginary quadratic fields.

Abstract

We show that any rational cubic hypersurface of dimension at least 33 defined over a number field $K$ vanishes on a $K$-rational projective line, reducing the previous lower bound of Wooley by two. For $K=\mathbb Q$ we can reduce the bound to 29. The main ingredients are a result on linear spaces on quadratic forms over suitable non-real quadratic field extensions, and recent work of Bernert and Hochfilzer on cubic forms over imaginary quadratic number fields for the rational case.