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

TL;DR

This paper investigates the structure of a specific order-3 collineation in projective geometry, analyzing its fixed points, lines, and planes under field reduction, to better understand the geometry of the Figueroa projective plane.

Contribution

It provides a detailed classification and counting of fixed elements of the projectivity in the field reduction setting, linking it to the geometry of the Figueroa plane.

Findings

Classification of fixed points, lines, and planes under the projectivity.

Count of fixed geometric elements in the reduced setting.

Insight into the structure of the Figueroa projective plane.

Abstract

Let $\phi$ be a collineation of $\mathrm{PG}\left(2, q^{3}\right)$ of order 3 which fixes a plane of order $q$ pointwise. The points of $\mathrm{PG}\left(2, q^{3}\right)$ can be partitioned into three types with respect to orbits of $\phi$ : fixed points; points $P$ with $P, P^{\phi}, P^{\phi^{2}}$ distinct and collinear; and points $P$ with $P, P^{\phi}, P^{\phi^{2}}$ 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.