Sharp Bertini theorem for plane curves over finite fields

Abstract

We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field ${\mathbb{F}}_q$ with $d\le q+1$, then there is an ${\mathbb{F}}_q$-line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ where $q=p^r$.

7 pages