Lines on Cubic Hypersurfaces Over Finite Fields
Résumé
We show that any smooth cubic hypersurface of dimension n defined over a finite field Fq contains a line defined over Fq in each of the following cases:
• n = 3 and q ≥ 11;
• n = 4 and q 6= 3;
• n ≥ 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)
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