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:
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\(n=3\) and \(q\ge 11\);
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\(n=4\), and \(q=2\) or \(q\ge 5\);
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\(n\ge 5\).
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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- 1.
In characteristic 2, the curves \( \Gamma _L\) and \( \widetilde{\Gamma }_L\) might not be nodal (see Lemma 4.13).
- 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.
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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Computer computations, https://github.com/alaface/CubLin
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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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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