An investigation of the tangent splash of a subplane of PG(2,q³)

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.