# Transverse lines to surfaces over finite fields

## TL;DR

The paper proves the existence of transverse lines over finite fields for smooth reflexive surfaces in projective 3-space, establishing bounds on the field size relative to the surface's degree.

## Contribution

It provides new bounds for the existence of transverse lines over finite fields, improving understanding of geometric configurations over finite fields.

## Key findings

- Existence of transverse lines for smooth reflexive surfaces when q ≥ c * degree(S).
- Existence of transverse lines for general surfaces when q ≥ degree(S)^2.
- Explicit bound c ≈ 1.7808 for reflexive surfaces.

## Abstract

We prove that if $S$ is a smooth reflexive surface in $\mathbb{P}^3$ defined over a finite field $\mathbb{F}_q$, then there exists an $\mathbb{F}_q$-line meeting $S$ transversely provided that $q\geq c\operatorname{deg}(S)$, where $c=\frac{3+\sqrt{17}}{4}\approx 1.7808$. Without the reflexivity hypothesis, we prove the existence of a transverse $\mathbb{F}_q$-line for $q\geq \operatorname{deg}(S)^2$.