Plane sections of Fermat surfaces over finite fields

Abstract

In this paper, we characterize all curves over $\mathbb{F}_q$ arising from a plane section $$ \mathcal{P} : X_3-e_0X_0-e_1X_1-e_2X_2 = 0 $$ of the Fermat surface $$ \mathcal{S} : X_0^d + X_1^d + X_2^d +X_3^d = 0, $$ where $q = p^{h} = 2d+1$ is a prime power, $p >3$, and $e_0, e_1, e_2 \in \mathbb{F}_q$. In particular, we will prove that any nonlinear component $\mathcal{G} \subseteq \mathcal{P} \cap \mathcal{S} $ is a smooth classical curve of degree $n\leqslant d$ attaining the St\"ohr-Voloch bound $$ \# \mathcal{G}(\mathbb{F}_q) \leqslant \frac{1}{2} n(n+q-1) - \frac{1}{2} i(n-2), $$ with $i \in \{0,1,2,3,n,3n\}$.