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Abstract
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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
.
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Keywords
Bruck–Bose representation, subplanes, exterior splash,
linear sets
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Mathematical Subject Classification 2010
Primary: 51E20
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Milestones
Received: 2 February 2015
Accepted: 2 March 2017
Published: 19 November 2018
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