Lines on cubic hypersurfaces over finite fields
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 $A(F(X))$.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2015
- DOI:
- 10.48550/arXiv.1510.05803
- arXiv:
- arXiv:1510.05803
- Bibcode:
- 2015arXiv151005803D
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14G15;
- 14J70;
- 14F20;
- 14G10
- E-Print:
- The example in Section 5.4.2 was corrected by Nicolas Addington. Kiran Kedlaya corrected an error in our proof of Theorem 5.2. He also kindly provided Proposition 5.5 and its proof. Daniel Bragg corrected an error in the statement of Theorem 4.12. Typo corrected in (5). This is a slightly expanded (and corrected) version of the text published in the Simons Publication Series