The planar projectivity of PG(2, $q^3$) of order 3 under field reduction

Abstract: Let $\phi$ be a collineation of $\mathrm{PG}\left(2, q<sup>{3}\right)$ of order 3 which fixes a plane of order $q$ pointwise. The points of $\mathrm{PG}\left(2, q<sup>{3}\right)$ can be partitioned into three types with respect to orbits of $\phi$ : fixed points; points $P$ with $P, P^{\phi}, P<sup>{\phi<sup>{2}}$ distinct and collinear; and points $P$ with $P, P^{\phi}, P<sup>{\phi<sup>{2}}$ not collinear. Under field reduction, the collineation $\phi$ corresponds to a projectivity $\sigma$ of $\operatorname{PG}(8, q)$ of order 3 . With respect to the field reduction and the orbits of $\sigma$, the points of $\mathrm{PG}(8, q)$ can be partitioned into six types. This article looks at the projectivity $\sigma$ in detail, and classifies and counts the fixed points, fixed lines and fixed planes. The motivation is to give a description of the lines of the Figueroa projective plane in the $\mathrm{PG}(8, q)$ field reduction setting.