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 $\mathcal{A}=\{(1,t,t^{2^h},t^{2^h+1})\mid t\in {\mathbb{F}}_q\}\cup \{(0,0,0,1)\}$, with $\gcd (n,h)=1$, be a $(q+1)$-arc and let $G_h\simeq \mathrm{PGL}(2,q)$ be the stabilizer of $\mathcal{A}$ 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 $\mathcal{L}$, 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 $\mathcal{L}$.