Covering plane curves over finite fields with lines

Abstract

We prove that, for every fixed $d\ge 2$ and $\epsilon >0$, the ${\mathbb{F}}_q$-points of a geometrically irreducible reflexive plane curve of degree $d$ over ${\mathbb{F}}_q$ can be covered by at most $\left(\frac{1}{d}+\epsilon \right)q$ many ${\mathbb{F}}_q$-lines, provided that $q$ is sufficiently large. The same conclusion holds for ${\mathbb{F}}_q$-irreducible curves that are not geometrically irreducible, since they have only $O(d^2)$ ${\mathbb{F}}_q$-points. In particular, for sufficiently large $q$, the bound applies to every ${\mathbb{F}}_q$-irreducible plane curve of degree $d$ when $\mathrm{char}({\mathbb{F}}_q)>d$. Our bound is asymptotically sharp. The proof combines Entin's Chebotarev density theorem for varieties over finite fields with the Pippenger--Spencer theorem on covers of nearly regular hypergraphs.

14 pages