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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2502.12495 |
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| _version_ | 1866913753175425024 |
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| author | Barwick, S. G. Hui, Alice M. W. Jackson, Wen-Ai |
| author_facet | Barwick, S. G. Hui, Alice M. W. Jackson, Wen-Ai |
| contents | Let $ϕ$ 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 $ϕ$ : fixed points; points $P$ with $P, P^ϕ, P^{ϕ^{2}}$ distinct and collinear; and points $P$ with $P, P^ϕ, P^{ϕ^{2}}$ not collinear. Under field reduction, the collineation $ϕ$ corresponds to a projectivity $σ$ of $\operatorname{PG}(8, q)$ of order 3 . With respect to the field reduction and the orbits of $σ$, the points of $\mathrm{PG}(8, q)$ can be partitioned into six types. This article looks at the projectivity $σ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The planar projectivity of PG(2, $q^3$) of order 3 under field reduction Barwick, S. G. Hui, Alice M. W. Jackson, Wen-Ai Combinatorics Let $ϕ$ 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 $ϕ$ : fixed points; points $P$ with $P, P^ϕ, P^{ϕ^{2}}$ distinct and collinear; and points $P$ with $P, P^ϕ, P^{ϕ^{2}}$ not collinear. Under field reduction, the collineation $ϕ$ corresponds to a projectivity $σ$ of $\operatorname{PG}(8, q)$ of order 3 . With respect to the field reduction and the orbits of $σ$, the points of $\mathrm{PG}(8, q)$ can be partitioned into six types. This article looks at the projectivity $σ$ 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. |
| title | The planar projectivity of PG(2, $q^3$) of order 3 under field reduction |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.12495 |