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

Abstract

Abstract Let π be a subplane of PG(2,q 3) of order q that is exterior to ℓ ∞. The exterior splash of π is the set of q 2+q+1 points on ℓ ∞ that lie on the extended lines 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 carrier conics of π in terms of the covers of the exterior splash of π. We also investigate properties of subplanes of order q with a common exterior splash, and study the intersection of two exterior splashes.