Abstract
Let be an order--subplane of that is exterior to . Then the exterior splash of is the set of points on that lie on an extended line of . Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry , and hyper-reguli in . We use the Bruck–Bose representation in to investigate the structure of , and the interaction between and its exterior splash. We show that the point set of corresponding to 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 , an exterior splash 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 . These transversal lines are used to characterise the carriers and the sublines of .
Citation
Susan G. Barwick. Wen-Ai Jackson. "The exterior splash in $\mathrm{PG}(6, q)$: transversals." Innov. Incidence Geom. Algebr. Topol. Comb. 17 (1) 1 - 24, 2019. https://doi.org/10.2140/iig.2019.17.1
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