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arXiv · 2307.09449

Rational lines on cubic hypersurfaces II

Julia Brandes·Rainer Dietmann·David B. Leep
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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.

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Julia Brandes, Rainer Dietmann, David B. Leep. 2025-11-21. Rational lines on cubic hypersurfaces II. https://arxiv.org/abs/2307.09449

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