Exterior splashes and linear sets of rank 3

Abstract

In PG(2, q 3 ), let π be a subplane of order q that is exterior to ℓ ∞ . The exterior splash of π is defined to be the set of q 2 + q + 1 points on ℓ ∞ that lie on a line of π. This article investigates properties of an exterior order-q-subplane and its exterior splash. We show that the following objects are projectively equivalent: exterior splashes, covers of the circle geometry CG(3, q), Sherk surfaces of size q 2 +q+1, and scattered linear sets of rank 3. We compare our construction of exterior splashes with the projection construction of a linear set. We give a geometric construction of the two different families of sublines in an exterior splash, and compare them to the known families of sublines in a scattered linear set of rank 3.

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

“…The group I = PGL(3, q 3 ) π,ℓ identifies two special points E 1 = ℓ ∩ m, E 2 = ℓ ∩ n on ℓ which are called the carriers of the exterior splash S of π onto ℓ. This is consistent with the definition of carriers of a circle geometry CG(3, q), see [7,Theorem 4.2]. The fixed points and fixed lines of I are used to define an important class of conics in π.…”

“…We first count x. By [7,Theorem 4.5], there are 2q 6 (q 3 − 1) exterior order-q-subplanes with the fixed exterior splash S. Further, an exterior order-q-subplane contains q 2 + q + 1 special conics. Hence x = 2q 6 (q 3 − 1)(q 2 + q + 1).…”

“…The exterior splash turns out to be a set rich in geometric structure. In particular, [7] showed that the sets of points in an exterior splash has arisen in many different situations, namely scattered linear sets of rank 3, Sherk surfaces of size q 2 + q + 1, covers of the circle geometry CG (3, q), and so hyper-reguli in PG (5, q). Properties of the exterior splash in the Bruck-Bose representation in PG(6, q) are studied in [8].…”

“…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.…”

On the equivalence of linear sets

“…It follows from [2, Proposition 3.1] that G is transitive on I and hence there is a unique scattered linear set of rank three in PG (1, q 3 ). An alternate proof can be obtained from [1], where the uniqueness of an exterior splash is proved, taking into account that the exterior splashes dealt with in that paper are precisely the scattered linear sets of rank three [11].…”