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

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

TL;DR

This work analyzes the partition of the point-set of ${PG}(2,q^3)$ induced by an order-$3$ planar collineation $\phi$ fixing an $_q$-plane $\mathcal{P}_{2,q}$. By fixing a Type $\textup{III}$ point $T$ and studying the subgroup ${\mathsf S_{\hbox{\tiny T}}}$, the authors classify the ${\mathsf S_{\hbox{\tiny T}}}$-orbits into seven categories, comprising the three fixed points, $q-1$ $T$-scattered-linear-sets on each side of the triangle $TT^{\phi}T^{\phi^2}$, and a large family of $\u00A0\mathbb{F}_q$-planes. They then explore the actions of $\phi$ and the involution $\mu$ on these orbits, identify fixed setwise planes $\pi_\lambda$ with $\lambda^3=1$, and relate points/lines via $\mu$ to construct and examine Fig-blocks used in the Figueroa plane construction. The paper further analyzes projection and splash maps from $\mathbb{F}_q$-planes to the line $m_T=T^{\phi}T^{\phi^2}$, and studies how these maps interact with the Fig-block structure, yielding a geometric characterization of Fig-blocks and insights into the distribution of $T$-planes among projection pencils. Overall, the results illuminate the geometry underlying the Figueroa plane ${FIG}(q^3)$ and suggest potential generalizations of the construction through a detailed understanding of $\mathsf{S_T}$-orbits and associated linear sets.

Abstract

Let $φ$ be a collineation of order 3 acting on $PG(2,q^3)$ whose fixed points are exactly an $\mathbb F_q$-plane $π_q$. Let $T$ be a point whose orbit under $φ$ is a triangle and let $S_G$ be the subgroup of $PGL(3,q^3)$ that fixes setwise the $\mathbb F_q$-plane $π_q$ and fixes setwise the line $T^φT^{φ^2}$. The point orbits of $S_G$ form a partition of the points of $PG(2,q^3)$ and consist of: the singletons $T,T^φ, T^{φ^2}$; scattered linear sets on the sides of the triangle $T T^φT^{φ^2}$; 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.

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

TL;DR

This work analyzes the partition of the point-set of induced by an order- planar collineation fixing an -plane . By fixing a Type point and studying the subgroup , the authors classify the -orbits into seven categories, comprising the three fixed points, -scattered-linear-sets on each side of the triangle , and a large family of -planes. They then explore the actions of and the involution on these orbits, identify fixed setwise planes with , and relate points/lines via to construct and examine Fig-blocks used in the Figueroa plane construction. The paper further analyzes projection and splash maps from -planes to the line , and studies how these maps interact with the Fig-block structure, yielding a geometric characterization of Fig-blocks and insights into the distribution of -planes among projection pencils. Overall, the results illuminate the geometry underlying the Figueroa plane and suggest potential generalizations of the construction through a detailed understanding of -orbits and associated linear sets.

Abstract

Let be a collineation of order 3 acting on whose fixed points are exactly an -plane . Let be a point whose orbit under is a triangle and let be the subgroup of that fixes setwise the -plane and fixes setwise the line . The point orbits of form a partition of the points of and consist of: the singletons ; scattered linear sets on the sides of the triangle ; and -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.

Paper Structure

This paper contains 8 sections, 16 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Coordinates
  3. The action of $\phi$ on elements of ${\textsl{orbits}}({\mathsf S_{\hbox{\tiny T}}})$
  4. The action of $\mu$ on elements of ${\textsl{orbits}}({\mathsf S_{\hbox{\tiny T}}})$
  5. Two more maps acting on $\mathbb{F}_{\space q}$-planes in ${\textsl{orbits}}({\mathsf S_{\hbox{\tiny T}}})$: projection and splash
  6. Projecting the Fig-block $\lDoubleBrace\space T \space\rDoubleBrace$
  7. Projection vertices and Figueroa lines
  8. Conclusion

Key Result

Theorem 2

The elements of ${\textsl{orbits}}({\mathsf S_{\hbox{\tiny T}}})$ consist of the following:

Figures (1)

  • Figure 1: Corollary \ref{['15cor']} describes the position of $T$-planes in relation to the $T$-slses. In this figure the Fig-block $\lDoubleBrace\space T \space\rDoubleBrace$ is shaded

Theorems & Definitions (30)

  • Definition 1
  • Theorem 2
  • proof
  • Lemma 4
  • proof
  • Corollary 5
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 20 more