Lines on cubic hypersurfaces over finite fields

We show that smooth cubic hypersurfaces of dimension $n$ defined over a finite field ${\mathbf{F}}_q$ contain a line defined over ${\mathbf{F}}_q$ in each of the following cases:

• $n=3$ and $q\ge 11$;
• $n=4$ and $q\mathrm{\ne }3$;
• $n\ge 5$. 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 $A(F(X))$.