# Tangent-filling plane curves over finite fields

## TL;DR

This paper investigates plane curves over finite fields with the property that their tangent lines at smooth rational points collectively cover all points in the projective plane over the finite field.

## Contribution

It introduces and analyzes a new class of plane curves over finite fields characterized by their tangent lines covering the entire projective plane.

## Key findings

- Identification of conditions for tangent lines to cover all points
- Construction of examples of such tangent-filling curves
- Insights into the geometric structure of these curves

## Abstract

We study plane curves over finite fields whose tangent lines at smooth $\mathbb{F}_q$-points together cover all the points of $\mathbb{P}^2(\mathbb{F}_q)$.