6 research outputs found
The tangent splash in \PG(6,q)
Let B be a subplane of PG(2,q^3) of order q that is tangent to .
Then the tangent splash of B is defined to be the set of q^2+1 points of
that lie on a line of B. In the Bruck-Bose representation of
PG(2,q^3) in PG(6,q), we investigate the interaction between the ruled surface
corresponding to B and the planes corresponding to the tangent splash of B. We
then give a geometric construction of the unique order--subplane determined
by a given tangent splash and a fixed order--subline.Comment: arXiv admin note: substantial text overlap with arXiv:1303.550
Subgeometries and linear sets on a projective line
We define the splash of a subgeometry on a projective line, extending the
definition of \cite{BaJa13} to general dimension and prove that a splash is
always a linear set. We also prove the converse: each linear set on a
projective line is the splash of some subgeometry. Therefore an alternative
description of linear sets on a projective line is obtained. We introduce the
notion of a club of rank , generalizing the definition from \cite{FaSz2006},
and show that clubs correspond to tangent splashes. We determine the condition
for a splash to be a scattered linear set and give a characterization of clubs,
or equivalently of tangent splashes. We also investigate the equivalence
problem for tangent splashes and determine a necessary and sufficient condition
for two tangent splashes to be (projectively) equivalent
Exterior splashes and linear sets of rank 3
In \PG(2,q^3), let be a subplane of order that is exterior to
\li. The exterior splash of is defined to be the set of
points on \li that lie on a line of . This article investigates
properties of an exterior \orsp\ and its exterior splash. We show that the
following objects are projectively equivalent: exterior splashes, covers of the
circle geometry , Sherk surfaces of size , and
\GF(q)-linear sets of rank 3 and size . 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 tangent splash in PG (6, q)
Abstract not availableS.G. Barwick, Wen-Ai Jackso