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arXiv · 2502.12495

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

S. G. Barwick·Alice M. W. Hui·Wen-Ai Jackson
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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.

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S. G. Barwick, Alice M. W. Hui, Wen-Ai Jackson. 2025-02-18. The planar projectivity of PG(2, $q^3$) of order 3 under field reduction. https://arxiv.org/abs/2502.12495

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