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

TL;DR

This paper studies the structure of exterior splashes in projective geometry, revealing their transversals and carriers using the Bruck-Bose representation in PG(6,q), and connects these to linear sets and hyper-reguli.

Contribution

It introduces a novel analysis of exterior splashes in PG(6,q), identifying their transversals and carriers through the Bruck-Bose model, and links these structures to known geometric configurations.

Findings

Each set of cover planes has three unique transversals in PG(6,q^3).

Transversals characterize the carriers and sublines of the exterior splash.

Exterior splashes are related to scattered linear sets, circle geometry covers, and hyper-reguli.

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 $\mathbb S$.