Rational lines on cubic hypersurfaces II
Number Theory
2025-11-25 v2
Abstract
We show that any rational cubic hypersurface of dimension at least 33 defined over a number field vanishes on a -rational projective line, reducing the previous lower bound of Wooley by two. For 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.
Cite
@article{arxiv.2307.09449,
title = {Rational lines on cubic hypersurfaces II},
author = {Julia Brandes and Rainer Dietmann and David B. Leep},
journal= {arXiv preprint arXiv:2307.09449},
year = {2025}
}