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

Abstract

Let π be an order-q-subplane of PG(2, q 3 ) that is exterior to ℓ ∞ . Then 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 in PG (5, q). In this article we use the Bruck-Bose representation in PG(6, q) to investigate the structure of π, and the interaction between π and its exterior splash. In PG(6, q), an exterior splash 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 S, and to characterise the sublines of S.

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

“…In [8], we show that in the cubic extension PG(5, q 3 ) of Σ ∞ ∼ = PG(5, q), each set S, T, C has a unique triple of conjugate transversal lines. That is, the lines g S , g q S , g q 2 S of PG(5, q 3 )\PG(5, q) meet every extended plane of S, and these are the only lines of PG(5, q 3 ) which meet every extended plane of S (in fact, these are the three transversal lines of the regular 2-spread S).…”

“…Firstly, we consider the structure of an exterior order-q-subplane π in PG (6, q). We showed in [8,Theorem 4.1] that [π] is the intersection of nine quadrics. We show in Theorem 3.11 that each of these nine quadrics contains the transversal lines of both the exterior splash S and the conic cover C of π. Secondly, in Theorem 3.12, we consider applications to replacement sets of PG(5, q).…”

“…This article relies heavily on properties of exterior splashes proved in [7,8]. In this section we detail the most important definitions and results that are needed in this article.…”

“…This theorem says that if we want to prove a result about exterior order-q-subplanes, or about exterior splashes, then we can without loss of generality prove it for a particular order-q-subplane. In [8], an order-q-subplane B which is exterior to ℓ ∞ is coordinatised, and B is used in many of the proofs in this article. The main points of this coordinatisation are as follows.…”

“…One cover is denoted T and is called the tangent cover of π (or the tangent cover of S with respect to π). The name follows from [8,Theorem 5.3] which shows that the planes of T are related to the tangent planes of [π]. The other cover is denoted C and is called the conic cover of π, it is related to certain conics of π, see Theorem 3.3.…”