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The exterior splash in $PG(6,q)$: Special conics

Let $π$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to $\ell_\infty$. 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 of $PG(5,q)$. In this article we use the Bruck-Bose representation in $PG(6,q)$ to give a geometric characterisation of special conics of $π$ in terms of the covers of the exterior splash of $π$. We also investigate properties of order-$q$-subplanes\ with a common exterior splash, and study the intersection of two exterior splashes.