Rational lines on cubic hypersurfaces

TL;DR

This paper proves that smooth projective cubic hypersurfaces of dimension at least 29 over the rationals contain rational lines, and extends the result to p-adic fields, improving previous bounds and showing rational points are generated from a single point.

Contribution

The authors establish the existence of rational lines on high-dimensional cubic hypersurfaces over the rationals and p-adic fields, refining earlier results and demonstrating generation of rational points from a single point.

Findings

Existence of rational lines on cubic hypersurfaces of dimension ≥29 over Q.

Extension of results to p-adic fields.

Rational points generated via secant and tangent methods from one point.

Abstract

We show that any smooth projective cubic hypersurface of dimension at least $29$ over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley. We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.