Lines on cubic hypersurfaces over finite fields

TL;DR

This paper proves the existence of rational lines on smooth cubic hypersurfaces over finite fields in various dimensions and sizes, and investigates the geometric properties of the associated line varieties, confirming the Tate conjecture in certain cases.

Contribution

It establishes conditions for the existence of lines over finite fields on cubic hypersurfaces and verifies the Tate conjecture for the line variety of a cubic threefold.

Findings

Existence of lines for specified dimensions and field sizes

Tate conjecture holds for the line variety of a cubic threefold

Information on Picard number and Albanese variety of the line variety

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 $A(F(X))$.