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Rational lines on cubic hypersurfaces

Part of: Arithmetic problems. Diophantine geometry Forms and linear algebraic groups Surfaces and higher-dimensional varieties Diophantine equations

Published online by Cambridge University Press:  24 April 2020

JULIA BRANDES  and
RAINER DIETMANN
JULIA BRANDES
Affiliation:
Mathematical Sciences, Chalmers Institute of Technology and University of Gothenburg, 41296Göteborg, Sweden. e-mail: brjulia@chalmers.se
RAINER DIETMANN
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham, TW20 OEX. e-mail: rainer.dietmann@rhul.ac.uk
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Abstract

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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.

MSC classification

Primary: 11D72: Equations in many variables
Secondary: 14G05: Rational points 11E76: Forms of degree higher than two 11D88: $p$-adic and power series fields 14J70: Hypersurfaces
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 1 , July 2021 , pp. 99 - 112
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Cambridge Philosophical Society 2020

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