arXiv subjects
Publications and source records attributed to S. G. Barwick.
A simple counting argument is used to show that for all $q$, an André hyper-regulus $\mathbb X$ in $PG(5,q)$ has exactly two switching sets. Moreover, there are exactly $2(q^2+q+1)$ planes in $PG(5,q)$ that meet every plane of $\mathbb X$ in a point, namely the planes in the switching sets.
In $PG(2,q^3)$, let $π$ be a subplane of order $q$ that is tangent to $\ell_infty$. The tangent splash of $π$ is defined to be the set of $q^2+1$ points on $\ell_infty$ 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 \cite{lavr10} to obtain properties of tangent splashes.
We consider a non-degenerate conic in $\PG(2,q^2)$, $q$ odd, that is tangent to $\ell_\infty$ and look at its structure in the Bruck-Bose representation in $\PG(4,q)$. We determine which combinatorial properties of this set of points in $\PG(4,q)$ are needed to reconstruct the conic in $\PG(2,q^2)$. That is, we define a set $\C$ in $\PG(4,q)$ with $q^2$ points that satisfies certain combinatorial properties. We then show that if $q\ge 7$, we can use $\C$ to construct a regular spread $§$ in the hyperplane at infinity of $\PG(4,q)$, and that $\C$ corresponds to a conic in the Desarguesian plane $¶(§)\cong\PG(2,q^2)$ constructed via the Bruck-Bose correspondence.
In this article we consider a set C of points in PG(4,q), q even, satisfying certain combinatorial properties with respect to the planes of PG(4,q). We show that there is a regular spread in the hyperplane at infinity, such that in the corresponding Bruck-Bose plane PG(2,q^2), the points corresponding to C form a translation hyperoval, and conversely.
Let B be a subplane of PG(2,q^3) of order q that is tangent to $\ell_\infty$. Then the tangent splash of B is defined to be the set of q^2+1 points of $\ell_\infty$ 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-$q$-subplane determined by a given tangent splash and a fixed order-$q$-subline.
In: S.G. Barwick and W.A. Jackson. Sublines and subplanes of PG(2,q^3) in the Bruck--Bose representation in PG(6,q). Finite Fields Th. App. 18 (2012) 93--107., the authors determine the representation of order-q-subplanes and order-q-sublines of PG(2,q^3) in the Bruck-Bose representation in PG(6,q). In particular, they showed that an order-q-subplane of PG(2,q^3) corresponds to a certain ruled surface in PG(6,q). In this article we show that the converse holds, namely that any ruled surface satisfying the required properties corresponds to a tangent order-q-subplane of PG(2,q^3).