On the geometry of a $(q + 1)$-arc of $\mathrm{PG}(3, q)$, q even

TL;DR

This paper investigates the geometric and group-theoretic properties of a specific $(q+1)$-arc in projective space $ ext{PG}(3,q)$ for even $q$, detailing orbit structures and incidence matrices under the stabilizer group.

Contribution

It explicitly determines the orbits of the stabilizer group on points, lines, and planes, and analyzes the incidence matrices, revealing dependence on trace functions for line distributions.

Findings

Orbit classifications on points, lines, and planes

Explicit point-plane incidence matrix under group action

Line point-orbit distribution depends on trace conditions

Abstract

In $\mathrm{PG}(3, q)$, $q = 2^n$, $n \ge 3$, let ${\cal A} = \{(1,t,t^{2^h},t^{2^h+1}) \mid t \in \mathbb{F}_q\} \cup \{(0,0,0,1)\}$, with $\mathrm{gcd}(n,h) = 1$, be a $(q+1)$-arc and let $G_h \simeq \mathrm{PGL}(2, q)$ be the stabilizer of $\cal A$ in $\mathrm{PGL}(4, q)$. The $G_h$-orbits on points, lines and planes of $\mathrm{PG}(3, q)$, together with the point-plane incidence matrix with respect to the $G_h$-orbits on points and planes of $\mathrm{PG}(3, q)$ are determined. The point-line incidence matrix with respect to the $G_1$-orbits on points and lines of $\mathrm{PG}(3, q)$ is also considered. In particular, for a line $\ell$ belonging to a given line $G_1$-orbits, say $\cal L$, the point $G_1$-orbit distribution of $\ell$ is either explicitly computed or it is shown to depend on the number of elements $x$ in $\mathbb{F}_q$ (or in a subset of $\mathbb{F}_q$) such that $\mathrm{Tr}_{q|2}(g(x)) = 0$, where $g$ is an $\mathbb{F}_q$-map determined by $\cal L$.