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Geometry Over Nonclosed Fields pp 19–51Cite as

Lines on Cubic Hypersurfaces Over Finite Fields

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Part of the book series: Simons Symposia ((SISY))

Abstract

We show that smooth cubic hypersurfaces of dimension n defined over a finite field \(\mathbf{F}_q\) contain a line defined over \(\mathbf{F}_q\) in each of the following cases:

  • \(n=3\) and \(q\ge 11\);

  • \(n=4\), and \(q=2\) or \(q\ge 5\);

  • \(n\ge 5\).

For a smooth cubic threefold X, the variety of lines contained in X is a smooth projective surface F(X) for which the Tate conjecture holds, and we obtain information about the Picard number of F(X) and the 5-dimensional principally polarized Albanese variety A(F(X)).

The second author was partially supported by Proyecto FONDECYT Regular N. 1150732 and Proyecto Anillo ACT 1415 PIA Conicyt.

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Notes

  1. 1.

    In characteristic 2, the curves \( \Gamma _L\) and \( \widetilde{\Gamma }_L\) might not be nodal (see Lemma 4.13).

  2. 2.

    A hypersurface singularity is of type \(A_j\) if it is, locally analytically, given by an equation \(x_1^{j+1}+x_2^2+\dots +x_{n+1}^2=0\). Type \(A_1\) is also called a node.

  3. 3.

    Among smooth cubics in \(\mathbf{P}^4_{\mathbf{F}_2}\) with no \(\mathbf{F}_2\)-lines, the computer found examples whose number of \(\mathbf{F}_2\)-points is any odd number between 3 and 13.

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Acknowledgements

This collaboration started after a mini-course given by the first author for the CIMPA School “II Latin American School of Algebraic Geometry and Applications” given in Cabo Frio, Brazil. We thank CIMPA for financial support and the organizers C. Araujo and S. Druel for making this event successful. Many thanks also to F. Charles, S. Elsenhans, B. van Geemen, F. Han, E. Howe, T. Katsura, Ch. Liedtke, and O. Wittenberg for useful correspondences and conversations. The computations made for this paper were done with the computer algebra programs Magma [6] and Sage.

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Authors and Affiliations

  1. Département Mathématiques et Applications, UMR CNRS 8553, PSL Research University, École Normale Supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

    Olivier Debarre

  2. Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Antonio Laface

  3. Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Université de Poitiers, Téléport 2 - BP 30179, 86962, Futuroscope Chasseneuil, France

    Xavier Roulleau

Authors
  1. Olivier Debarre
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  2. Antonio Laface
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  3. Xavier Roulleau
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Corresponding author

Correspondence to Olivier Debarre .

Editor information

Editors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, New York, USA

    Fedor Bogomolov

  2. Department of Mathematics, Brown University, Providence, Rhode Island, USA

    Brendan Hassett

  3. Courant Institute of Mathematical Sciences and Simons Foundation, New York University, New York, New York, USA

    Yuri Tschinkel

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Debarre, O., Laface, A., Roulleau, X. (2017). Lines on Cubic Hypersurfaces Over Finite Fields. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds) Geometry Over Nonclosed Fields. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-49763-1_2

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