Abstract

Let $\pi$ be an order-$q$-subplane of $PG\left(2,q^3\right)$ that is exterior to ${\ell }_{\infty }$. Then the exterior splash of $\pi$ is the set of $q^2+q+1$ points on ${\ell }_{\infty }$ that lie on an extended line of $\pi$. Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry $CG\left(3,q\right)$, and hyper-reguli in $PG\left(5,q\right)$. We use the Bruck–Bose representation in $PG\left(6,q\right)$ to investigate the structure of $\pi$, and the interaction between $\pi$ and its exterior splash. We show that the point set of $PG\left(6,q\right)$ corresponding to $\pi$ is the intersection of nine quadrics, and that there is a unique tangent plane at each point, namely the intersection of the tangent spaces of the nine quadrics. In $PG\left(6,q\right)$, an exterior splash $\mathbb{S}$ has two sets of cover planes (which are hyper-reguli) and we show that each set has three unique transversal lines in the cubic extension $PG\left(6,q^3\right)$. These transversal lines are used to characterise the carriers and the sublines of $\mathbb{S}$.

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Mathematical Subject Classification 2010

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