Chapitre D'ouvrage Année : 2017

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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Dates et versions

hal-03807436 , version 1 (17-02-2024)

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Identifiants

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Olivier Debarre, Antonio Laface, Xavier Roulleau. Lines on Cubic Hypersurfaces Over Finite Fields. Fedor Bogomolov; Brendan Hassett; Yuri Tschinkel. Geometry Over Nonclosed Fields, Springer International Publishing, pp.19-51, 2017, Simons Symposia, 978-3-319-49762-4. ⟨10.1007/978-3-319-49763-1_2⟩. ⟨hal-03807436⟩
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