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arXiv · 2112.12452

Two disguises of the linear representation of a subgeometry

Lins Denaux
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Abstract

Let $\text{PG}(n,q)$ be the Desarguesian projective space of dimension $n$ over the finite field of order $q$. The \emph{linear representation} of a point set $\mathcal{K}$ in a hyperplane at infinity of $\text{PG}(n,q)$ is the point-line geometry consisting of the affine points of $\text{PG}(n,q)$, together with the union of the parallel classes of affine lines corresponding to the points of $\mathcal{K}$. This type of point-line geometry has been widely investigated in the literature. Curiously, if $\mathcal{K}$ is a subgeometry, two disguises of its linear representation occur in two separate works. In this short note, we give an explicit isomorphism between these two disguises by making use of field reduction.

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Lins Denaux. 2021-12-23. Two disguises of the linear representation of a subgeometry. https://arxiv.org/abs/2112.12452

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