Hostname: page-component-cb9f654ff-pvkqz Total loading time: 0 Render date: 2025-09-02T00:56:17.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Rational lines on cubic hypersurfaces

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

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, 41296 Göteborg, 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

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

References

REFERENCES

Birch, B. J.. Homogeneous forms of odd degree in a large number of variables. Mathematika 4 (1957), 102–105.CrossRefGoogle Scholar
Brandes, J.. A note on p-adic solubility for forms in many variables. Bull. London Math. Soc. 47 (2015), 501–508.CrossRefGoogle Scholar
Browning, T. D., Dietmann, R. and Heath–Brown, D. R.. Rational points on intersections of cubic and quadric hypersurfaces. J. Inst. Math. Jussieu 14 (2015), no. 4, 703–749.10.1017/S1474748014000127CrossRefGoogle Scholar
Brüdern, J., Dietmann, R., Liu, J. and Wooley, T. D.. A Birch–Goldbach theorem. Arch. Math. 94 (2010), 53–58.CrossRefGoogle Scholar
Cook, B. and Magyar, Á.. Diophantine equations in the primes. Invent. Math. 198 (2014), no. 3, 701–737.CrossRefGoogle Scholar
Debarre, O., Laface, A. and Roulleau, X.. Lines on cubic hypersurfaces over finite fields. Geometry over nonclosed fields, Simons Symp. (Springer, Cham, 2017), pp. 19–51.Google Scholar
Dietmann, R.. Systems of cubic forms. J. London Math. Soc. 77 (2008), 666–686.CrossRefGoogle Scholar
Dietmann, R. and Wooley, T. D.. Pairs of cubic forms in many variables. Acta Arith. 110 (2003), no. 2, 125–140.CrossRefGoogle Scholar
Harris, J.. Algebraic geometry. A first course. Graduate Texts in Mathematics, vol. 133 (Springer-Verlag, New York, 1992).CrossRefGoogle Scholar
Heath-Brown, D. R.. Cubic forms in ten variables. Proc. London Math. Soc. (3) 47 (1983), 225–257.CrossRefGoogle Scholar
Hua, L.–K.. Additive theory of prime numbers. Amer. Math. Soc. (Providence, Rhode Island, 1965).Google Scholar
Kumchev, A. V. and Wooley, T. D.. On the Waring Goldbach problem for seventh and higher powers. Monatsh. Math. 183 (2017), no. 2, 303–310.10.1007/s00605-016-0936-7CrossRefGoogle Scholar
Lewis, D. J.. Cubic homogeneous polynomials over p-adic number fields. Ann. of Math. (2) 56 (1952), 473–478.CrossRefGoogle Scholar
Liu, J.. Integral points on quadrics with prime coordinates. Monatsh. Math. 164 (2011), no. 4, 439–465.CrossRefGoogle Scholar
Manin, Yu. I.. Cubic Forms. second ed., (North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986), Algebra, geometry, arithmetic. Translated from the Russian by M. Hazewinkel.Google Scholar
Mordell, L. J.. On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc. Cam. Phil. Soc. 21 (1922), 179–192.Google Scholar
Papanikolopoulos, S. and Siksek, S.. A Mordell-Weil theorem for cubic hypersurfaces of high dimensions, Algebra Number Theory 11 (2017), no. 8, 1953–1965.CrossRefGoogle Scholar
Serre, J.-P.. A Course in Arithmetic, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French, Graduate Texts in Math., No. 7.10.1007/978-1-4684-9884-4CrossRefGoogle Scholar
Skinner, C. M.. Forms over number fields and weak approximation. Compositio Math. 106 (1997), no. 1, 11–29.CrossRefGoogle Scholar
Viro, O.. Mutual position of hypersurfaces in projective space. Geometry of differential equations. Amer. Math. Soc. Transl. Ser. 2, vol. 186. (Amer. Math. Soc., Providence, RI, 1998), pp. 161–176.CrossRefGoogle Scholar
Wooley, T. D.. Forms in many variables, Analytic Number Theory: Proceedings of the 39th Taniguchi International Symposium, Kyoto, May, 1996 (Motohashi, Y., ed.). London Math. Soc. Lecture Note Ser. 247(Cambridge University Press, 1997), pp. 361–376.CrossRefGoogle Scholar
Wooley, T. D.. Linear spaces on cubic hypersurfaces and pairs of homogeneous cubic equations. Bull. London Math. Soc. 29 (1997), 556–562.CrossRefGoogle Scholar
Wooley, T. D.. An explicit version of Birch’s theorem. Acta Arith. 85 (1998), 79–96.10.4064/aa-85-1-79-96CrossRefGoogle Scholar
Zahid, J.. Simultaneous zeros of a cubic and quadratic form. J. Lond. Math. Soc. (2) 84 (2011), no. 3, 612–630.CrossRefGoogle Scholar