An investigation of the tangent splash of a subplane of $$\mathrm{PG}(2,q^3)$$ PG ( 2 , q 3 )

Abstract

In PG(2, q 3 ), let π be a subplane of order q that is tangent to ∞ . The tangent splash of π is defined to be the set of q 2 + 1 points on ∞ that lie on a line of π. This article investigates properties of the tangent splash. We show that all tangent splashes are projectively equivalent, investigate sublines contained in a tangent splash, and consider the structure of a tangent splash in the Bruck-Bose representation of PG(2, q 3 ) in PG(6, q). We show that a tangent splash of PG(1, q 3 ) is a GF(q)-linear set of rank 3 and size q 2 + 1; this allows us to use results about linear sets from [17] to obtain properties of tangent splashes. arXiv:1303.5509v2 [math.CO] 7 Apr 2014 2 The Bruck-Bose representation of PG(2, q 3 ) in PG(6, q) The Bruck-Bose representationWe begin by describing the Bruck-Bose representation of PG(2, q 3 ) in PG(6, q), and introduce the notation we will use.A 2-spread of PG(5, q) is a set of q 3 + 1 planes that partition PG(5, q). A 2-regulus of

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

“…In Section 4.3, we start with an exterior splash S and appropriate order-q-subline b, and give a geometric construction in the PG(6, q) setting of the two order-q-subplanes that share S and b. Note that this is analogous to the construction in [6] which gives a similar construction in the case of an order-q-subplane tangent to ℓ ∞ with a fixed tangent splash on ℓ ∞ .…”

Subgeometries and linear sets on a projective line

“…This work is motivated by the link between splashes and linear sets on a projective line. The concept of a splash of a subplane, although quite a natural geometric object to consider, has been studied only recently, see [1,2]. This paper extends the definition of a splash from subplanes to subgeometries of order q in higher dimensional projective spaces, and from cubic to general extension fields.…”

“…Precisely, if we denote the set of hyperplanes of a projective space π by H(π), and U denotes the extension of a subspace U of the subgeometry π 0 to a subspace of π, then we obtain the set of points {l ∞ ∩H : H ∈ H(π 0 )}. These sets have been studied in [1] and [2] for Desarguesian planes and cubic extensions, i.e. for a subplane π 0 ∼ = PG (2, q) in π ∼ = PG(2, q 3 ), where such a set is called the splash of π 0 on l ∞ .…”

“…Moreover, this generalization leads to a new interpretation of linear sets on a projective line. The equivalence stated in Theorem 3.1 may turn out useful in investigating linear sets, for instance by linking them to certain ruled surfaces in affine (2n)-dimensional spaces over F q , relying on results from [1,2]. Linear sets and field reduction have played an important role in the construction and characterization of many objects in finite geometry in recent years.…”

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

“…However, an order-q-subplane of PG(2, q 3 ) that is exterior to ℓ ∞ forms a complex structure in PG (6, q), and one of our original motivations was to investigate this structure in PG (6, q). As part of this investigation, we studied the splash of an order-q-subplane, and found this to be a structure rich in detail, and related to many other structures in geometry, see [4,5,6,18]. In this article we study the structure of an exterior order-q-subplane in PG (6, q), as well as the interplay between this structure and the associated exterior splash.…”