# Fragments of plane filling curves of degree $q+2$ over the finite field of $q$ elements, and of affine-plane filling curves of degree $q+1$

## TL;DR

This paper classifies certain plane filling curves over finite fields of specific degrees using matrix representations, providing insights into their degenerations and affine-plane analogs.

## Contribution

It introduces a matrix-based classification of nonsingular plane curves of degree q+2 over finite fields and explores affine-plane filling curves of degree q+1.

## Key findings

- Classification of degenerations via matrix representation
- Representation of curves by 3x3 matrices over finite fields
- Analysis of affine-plane filling curves of degree q+1

## Abstract

Nonsingular plane curves over a finite field $\mathbb{F}_q$ of degree $q+2$ passing through all the $\mathbb{F}_q$-points of the plane admita representation by $3\times 3$ matrices over $\mathbb{F}_q$. We classify their degenerations by means of the matrix representation, and also discuss the similar problem for the affine-plane filling curves of degree $q+1$.