The partition of $PG(2,q^3)$ arising from an order 3 planar collineation

Abstract

Let $\phi$ be a collineation of order 3 acting on $PG(2,q^3)$ whose fixed points are exactly an $\mathbb F_q$-plane $\pi_q$. Let $T$ be a point whose orbit under $\phi$ is a triangle and let $S_G$ be the subgroup of $PGL(3,q^3)$ that fixes setwise the $\mathbb F_q$-plane $\pi_q$ and fixes setwise the line $T^\phi T^{\phi^2}$. The point orbits of $S_G$ form a partition of the points of $PG(2,q^3)$ and consist of: the singletons $T,T^\phi, T^{\phi^2}$; scattered linear sets on the sides of the triangle $T T^\phi T^{\phi^2}$; and $\mathbb F_q$-planes. This article studies the structure of this partition, looking at maps that permute elements of the partition. The motivation in studying this partition lies in its application to the construction of the Figueroa projective plane, and the article concludes with a characterisation in this setting.