#### Vol. 17, No. 1, 2019

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The exterior splash in $\mathrm{PG}(6, q)$: transversals

### Susan G. Barwick and Wen-Ai Jackson

Vol. 17 (2019), No. 1, 1–24
##### 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 $\mathbb{S}$.

##### Keywords
Bruck–Bose representation, subplanes, exterior splash, linear sets
Primary: 51E20