Plane sections of Fermat surfaces over finite fields

TL;DR

This paper characterizes all plane sections of Fermat surfaces over finite fields, identifying the structure and bounds of nonlinear components, and providing explicit bounds on the number of rational points.

Contribution

It provides a complete characterization of curves from plane sections of Fermat surfaces over finite fields, including bounds on their rational points and smoothness properties.

Findings

Nonlinear components are smooth classical curves of degree ≤ d.

Bounds on the number of rational points on these curves are established.

Explicit classification of components based on intersection properties.

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 $i \in \{0,1,2,3,n,3n\}$.