Lines on cubic hypersurfaces over finite fields

Olivier Debarre; Antonio Laface; Xavier Roulleau

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arXiv · 1510.05803

Lines on cubic hypersurfaces over finite fields

Olivier Debarre·Antonio Laface·Xavier Roulleau

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Abstract

We show that smooth cubic hypersurfaces of dimension $n$ defined over a finite field ${\bf F}_q$ contain a line defined over ${\bf F}_q$ in each of the following cases: - $n=3$ and $q\ge 11$; - $n=4$ and $q\ne 3$; - $n\ge 5$. 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 its 5-dimensional principally polarized Albanese variety $A(F(X))$.

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Olivier Debarre, Antonio Laface, Xavier Roulleau. 2015-10-20. Lines on cubic hypersurfaces over finite fields. https://arxiv.org/abs/1510.05803

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