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

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.