Curves on Frobenius classical surfaces in $\mathbb{P}^3$ over finite fields

TL;DR

This paper establishes an improved upper bound on the number of rational points on irreducible space curves of a given degree lying on Frobenius classical surfaces over finite fields, advancing understanding of their arithmetic properties.

Contribution

It introduces a new upper bound for rational points on space curves on Frobenius classical surfaces, surpassing previous bounds in certain parameter ranges.

Findings

New upper bound on rational points for space curves on Frobenius classical surfaces.

Improved bounds compared to existing results in specific degree and field size ranges.

Enhanced understanding of the arithmetic properties of curves on algebraic surfaces.

Abstract

In this paper we give an upper bound on the number of rational points on an irreducible curve $C$ of degree $\delta$ defined over a finite field $\mathbb{F}_q$ lying on a Frobenius classical surface $S$ embedded in $\mathbb{P}^3$. This leads us to investigate arithmetic properties of curves lying on surfaces. In a certain range of $\delta$ and $q$, our result improves all other known bounds in the context of space curves.