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

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.