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

Abstract

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

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

“…The cover denoted C in the following lemma is called conic cover of S with respect to B, or the conic cover of [B]. The nomenclature for tangent covers comes from Theorem 5.3 which shows that the tangent planes T P of [B] meet Σ ∞ in lines that lie in distinct planes of the cover T. The nomenclature for the conic cover comes from [7] which shows that the planes in the cover C are related to the (B, ℓ ∞ )-special conics in B.…”

“…Consider the homography β defined in (7), acting on PG(5, q 3 ). The proof of Theorem 6.2 shows that β maps g S to g T , and maps g T to g C .…”

“…The notion of special conics and special twisted cubics is closely tied with this interplay. This theme is continued in [7] where the authors show that (π, ℓ ∞ )-special conics of π in PG(2, q 3 ) correspond to C-special conics in PG(6, q).…”

“…We use the coordinatisation in PG(5, q) of the exterior splash S of B and the two covers T, C given in Lemma 5.1. The homography β of PG(5, q) defined in (7) can be regarded as a homography of PG(5, q 3 ). By the proof of Lemma 5.1, β maps the spread plane…”

A characterisation of F_q-conics of PG(2,q^3)

“…The representation of F q -subplanes of PG(2, q 3 ) exterior to ℓ ∞ was investigated in [5,6,7]. The representations of F q -sublines, and tangent and secant F q -subplanes of PG(2, q 3 ) in PG(6, q) were determined in [3,4].…”