The exterior splash in PG(6,q): Transversals

Let $π$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to $\ell_\infty$. Then the exterior splash of $π$ is the set of $q^2+q+1$ points on $\ell_\infty$ that lie on an extended line of $π$. Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry $CG(3,q)$, and hyper-reguli in $PG(5,q)$. In this article we use the Bruck-Bose representation in $PG(6,q)$ to investigate the structure of $π$, and the interaction between $π$ and its exterior splash. In $PG(6,q)$, an exterior splash $\mathbb S$ has two sets of cover planes (which are hyper-reguli) and we show that each set has three unique transversals lines in the cubic extension $PG(6,q^3)$. These transversal lines are used to characterise the carriers of $\mathbb S$, and to characterise the sublines of $\mathbb S$.

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