Rational lines on cubic hypersurfaces
Number Theory
2021-07-01 v2
Abstract
We show that any smooth projective cubic hypersurface of dimension at least 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.
Cite
@article{arxiv.1809.08041,
title = {Rational lines on cubic hypersurfaces},
author = {Julia Brandes and Rainer Dietmann},
journal= {arXiv preprint arXiv:1809.08041},
year = {2021}
}
Comments
An oversight in Lemma 3.1 as well as a few typos have been corrected