A bound on the number of points of a curve in projective space over a finite field

TL;DR

This paper establishes an upper bound on the number of rational points on a nondegenerate irreducible curve in projective space over a finite field, extending known bounds to higher dimensions.

Contribution

It generalizes Sziklai's bound from plane curves to curves in higher-dimensional projective spaces over finite fields.

Findings

Proves that $N_q(C) \,\leq\; (d-1)q + 1$ for curves in ${\mathbb P}^r$ with $r \geq 3$

Extends classical bounds on rational points to higher-dimensional projective spaces

Provides a key inequality for algebraic geometry over finite fields

Abstract

For a nondegenerate irreducible curve $C$ of degree $d$ in ${\Bbb P}^r$ over ${\Bbb F}_q$ with $r \geq 3$, we prove that the number $N_q(C)$ of ${\Bbb F}_q$-points of $C$ satisfies the inequality $N_q(C) \leq (d-1)q +1$, which is known as Sziklai's bound if $r=2$.