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

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 $\cal L$.