Number of rational branches of a plane singular curve over a finite field

TL;DR

This paper investigates the number of rational points on a non-singular model of a plane singular curve over a finite field, focusing on curves with two singularities whose multiplicities sum to the degree, providing bounds and characterizations.

Contribution

It characterizes plane models with two singularities whose multiplicities sum to the degree and constructs examples attaining certain bounds for finite fields with characteristic greater than 3.

Findings

Complete characterization of such plane models.

Construction of curves attaining the bounds for p > 3.

Provides bounds for the number of rational points on the non-singular model.

Abstract

Let $\mathcal{F}$ be a plane singular curve defined over a finite field $\mathbb{F}_q$. The linear system of plane curves of a given degree passing through the singularities of $\cF$ provides potentially good bounds for the number of points on a non-singular model of $\mathcal{F}$. In this note, the case of a curve with two singularities such that the sum of their multiplicities is precisely the degree of the curve is investigated in more depth. In particular, such plane models are completely characterized, and for $p > 3$, a curve of this type attaining one of the obtained bounds is presented.