Vol. 17, No. 1, 2019

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The exterior splash in $\mathrm{PG}(6, q)$: transversals

Susan G. Barwick and Wen-Ai Jackson

Vol. 17 (2019), No. 1, 1–24

Let π be an order-q-subplane of PG(2,q3) that is exterior to . Then the exterior splash of π is the set of q2 + q + 1 points on 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). 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. We show that the point set of PG(6,q) corresponding to π is the intersection of nine quadrics, and that there is a unique tangent plane at each point, namely the intersection of the tangent spaces of the nine quadrics. 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 transversal lines in the cubic extension PG(6,q3). These transversal lines are used to characterise the carriers and the sublines of S.

Bruck–Bose representation, subplanes, exterior splash, linear sets
Mathematical Subject Classification 2010
Primary: 51E20
Received: 2 February 2015
Accepted: 2 March 2017
Published: 19 November 2018
Susan G. Barwick
School of Mathematical Sciences
University of Adelaide
Wen-Ai Jackson
School of Mathematical Sciences
University of Adelaide