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arXiv · 1303.5509

An investigation of the tangent splash of a subplane of PG(2,q^3)

S. G. Barwick·Wen-Ai Jackson
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Abstract

In $PG(2,q^3)$, let $\pi$ be a subplane of order $q$ that is tangent to $\ell_infty$. The tangent splash of $\pi$ is defined to be the set of $q^2+1$ points on $\ell_infty$ that lie on a line of $\pi$. This article investigates properties of the tangent splash. We show that all tangent splashes are projectively equivalent, investigate sublines contained in a tangent splash, and consider the structure of a tangent splash in the Bruck-Bose representation of $PG(2,q^3)$ in $PG(6,q)$. We show that a tangent splash of $PG(1,q^3)$ is a $GF(q)$-linear set of rank 3 and size $q^2+1$; this allows us to use results about linear sets from \cite{lavr10} to obtain properties of tangent splashes.

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S. G. Barwick, Wen-Ai Jackson. 2013-03-22. An investigation of the tangent splash of a subplane of PG(2,q^3). https://arxiv.org/abs/1303.5509

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