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 $$\mathcal{P}:X_3-e_0{X}_0-e_1{X}_1-e_2{X}_2=0$$ 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 $n⩽d$ attaining the Stöhr-Voloch bound $$\#\mathcal{G}({\mathbb{F}}_q)⩽\frac{1}{2}n(n+q-1)-\frac{1}{2}i(n-2),$$ with $i\in \{0,1,2,3,n,3n\}$.