Rational linear subspaces of hypersurfaces over finite fields

Abstract

Let $X\subset {\mathbb{P}}^n$ be a hypersurface of degree $d$ defined over a finite field of characteristic $p>0$. We prove that if $n\ge r+\left(\genfrac{}{}{0ex}{}{d+r}{r+1}\right)$, then $X$ contains a rational $r$-plane. We prove better bounds when $X$ is smooth and $p$ is sufficiently large. We also present experimental data regarding the existence of rational lines on cubic threefolds over ${\mathbb{F}}_7$, ${\mathbb{F}}_8$, and ${\mathbb{F}}_9$. In particular, we construct an example of a smooth cubic threefold over ${\mathbb{F}}_7$ with exactly $8$ rational lines. It remains an open question whether smooth cubic threefolds over ${\mathbb{F}}_7$, ${\mathbb{F}}_8$, and ${\mathbb{F}}_9$ always contain a rational line.

8 pages. v2: Rewrote introduction, simplified proof of Theorem 1.2