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

TL;DR

This paper characterizes special conics in exterior order-$q$-subplanes of $PG(2,q^3)$ using the Bruck-Bose representation in $PG(6,q)$, linking geometric structures like covers and splashes.

Contribution

It provides a geometric characterization of special conics via covers of exterior splashes in the Bruck-Bose model, and explores properties of subplanes sharing splashes.

Findings

Characterization of special conics in terms of covers

Properties of subplanes with common exterior splashes

Analysis of intersections between exterior splashes

Abstract

Let $\pi$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to $\ell_\infty$. 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 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 $\pi$ in terms of the covers of the exterior splash of $\pi$. We also investigate properties of order-$q$-subplanes\ with a common exterior splash, and study the intersection of two exterior splashes.