Sharp Bertini theorem for plane curves over finite fields

TL;DR

This paper proves that certain smooth plane curves over finite fields have lines intersecting them transversely, extending classical results to specific non-reflexive cases and providing new geometric insights.

Contribution

It establishes a sharp Bertini theorem for plane curves over finite fields, including reflexive and specific non-reflexive cases, enhancing understanding of curve-line intersections.

Findings

Existence of transverse lines for reflexive curves with degree ≤ q+1

Extension of the result to non-reflexive curves of degree p+1 and 2p+1

Improved understanding of geometric properties of 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 $d\leq q+1$, 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 $q=p^{r}$.