Curves on Frobenius classical surfaces in $\mathbb{P}^{3}$ over finite fields
Volume 205 / 2022
Acta Arithmetica 205 (2022), 323-340
MSC: Primary 11G20; Secondary 14G05, 14H50, 14J70.
DOI: 10.4064/aa211118-12-9
Published online: 17 October 2022
Abstract
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.
