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

Abstract

Let $\pi$ be an order-$q$-subplane of $PG(2,q^3)$ 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 $CG(3,q)$, 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 $\pi$, and the interaction between $\pi$ 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 $\mathbb{S}$.

References in corpus (1)

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