arXiv subjects
Publications and source records attributed to S. G. Barwick.
This article studies the intersection of two linear sets of rank 3 of PG(1,q^3). In particular, all possible intersection sizes are determined, the structure of the largest intersection sets is described, and existence results are given.
André planes of order $q^t$ can be constructed from the Desarguesian plane $PG(2,q^t)$ using a process that involves replacing André nets. This article constructs ovals in André planes of order $q^t$, $q$ even and $t$ prime that are inherited from conics of the Desarguesian plane $PG(2,q^t)$. Further, it is shown that certain conics do not inherit to arcs under the process of André replacement.
The process of deriving the Desarguesian plane $PG(2,q^2)$ to get the Hall plane is well known, and the problem of when a conic in $PG(2,q^2)$ inherits to an arc in the Hall plane has been solved. In this article we look at the generalisation of replacing an André net of $PG(2,q^t)$, $t\geq 3$ to construct an André plane of order $q^t$. This article looks at the case where $q$ is odd and $t$ is prime, and determines when a conic in $PG(2,q^t)$ inherits to an arc in an André plane. Further, the number of arcs in an André plane that are inherited in this way is enumerated.
Let $ϕ$ be a collineation of order 3 acting on $PG(2,q^3)$ whose fixed points are exactly an $\mathbb F_q$-plane $π_q$. Let $T$ be a point whose orbit under $ϕ$ is a triangle and let $S_G$ be the subgroup of $PGL(3,q^3)$ that fixes setwise the $\mathbb F_q$-plane $π_q$ and fixes setwise the line $T^ϕT^{ϕ^2}$. The point orbits of $S_G$ form a partition of the points of $PG(2,q^3)$ and consist of: the singletons $T,T^ϕ, T^{ϕ^2}$; scattered linear sets on the sides of the triangle $T T^ϕT^{ϕ^2}$; and $\mathbb F_q$-planes. This article studies the structure of this partition, looking at maps that permute elements of the partition. The motivation in studying this partition lies in its application to the construction of the Figueroa projective plane, and the article concludes with a characterisation in this setting.
Let $\phi$ be a collineation of $\mathrm{PG}\left(2, q^{3}\right)$ of order 3 which fixes a plane of order $q$ pointwise. The points of $\mathrm{PG}\left(2, q^{3}\right)$ can be partitioned into three types with respect to orbits of $\phi$ : fixed points; points $P$ with $P, P^{\phi}, P^{\phi^{2}}$ distinct and collinear; and points $P$ with $P, P^{\phi}, P^{\phi^{2}}$ not collinear. Under field reduction, the collineation $\phi$ corresponds to a projectivity $\sigma$ of $\operatorname{PG}(8, q)$ of order 3 . With respect to the field reduction and the orbits of $\sigma$, the points of $\mathrm{PG}(8, q)$ can be partitioned into six types. This article looks at the projectivity $\sigma$ in detail, and classifies and counts the fixed points, fixed lines and fixed planes. The motivation is to give a description of the lines of the Figueroa projective plane in the $\mathrm{PG}(8, q)$ field reduction setting.
This article considers an F_q-conic contained in an F_q-subplane of PG(2,q^3), and shows that it corresponds to a normal rational curve in the Bruck-Bose representation in PG(6,q). This article then characterises which normal rational curves of PG(6,q) correspond via the Bruck-Bose representation to F_q-conics of PG(2,q^3). The normal rational curves of interest are called 3-special, which relates to how the extension of the normal rational curve meets the transversal lines of the regular 2-spread of the Bruck-Bose representation. This article uses geometric arguments that exploit the interaction between the Bruck-Bose representation of PG(2,q^3) in PG(6,q), and the Bose representation of PG(2,q^3) in PG(8,q).
In this article we look at the geometric structure of the feet of an orthogonal Buekenhout-Metz unital U in PG(2,q^2). We show that the feet of each point form a set of type (0,1,2,4). Further, we discuss the structure of any 4-secants, and determine exactly when the feet form an arc.
This article looks at the Bose representation of $PG(2,q^3)$ as a 2-spread of $PG(8,q)$. It is shown that an $\mathbb F_q$-subline of $PG(2,q^3)$ corresponds to a 2-regulus, and an $\mathbb F_q$-subplane corresponds to a Segre variety $S_{2;2}$. Moreover, the extension of these varieties to $PG(8,q^3)$ and $PG(8,q^6)$ is determined. These are used to determine the structure of an $\mathbb F_q$-conic of $PG(2,q^3)$ in the Bose representation in $PG(8,q)$.
This article looks at subconics of order $q$ of $PG(2,q^2)$ and characterizes them in the Bruck-Bose representation in $PG(4,q)$. In common with other objects in the Bruck-Bose representation, the characterisation uses the transversals of the regular line spread $S$ associated with the Bruck-Bose representation.By working in the Bose representation of $PG(2,q^2)$ in $PG(5,q)$, we give a geometric explanation as to why the transversals of the regular spread $S$ are intrinsic to the characterisation of varieties of $PG(2,q^2)$.
This article studies conics and subconics of $PG(2,q^2)$ and their representation in the André/Bruck-Bose setting in $PG(4,q)$. In particular, we investigate their relationship with the transversal lines of the regular spread. The main result is to show that a conic in a tangent Baer subplane of $PG(2,q^2)$ corresponds in $PG(4,q)$ to a normal rational curve that meets the transversal lines of the regular spread. Conversely, every 3 and 4-dimensional normal rational curve in $PG(4,q)$ that meets the transversal lines of the regular spread corresponds to a conic in a tangent Baer subplane of $PG(2,q^2)$.
Let $E$ be a set of solids (hyperplanes) in $PG(4,q)$, $q$ even, $q>2$, such that every point of $PG(4,q)$ lies in either $0$, $\frac12q^3$ or $\frac12(q^3-q^2)$ solids of $E$, and every plane of $PG(4,q)$ lies in either $0$, $\frac12q$ or $q$ solids of $E$. This article shows that $E$ is either the set of solids that are disjoint from a hyperoval, or the set of solids that meet a non-singular quadric $Q(4,q)$ in an elliptic quadric.
Let $K$ be a set of $q^2+2q+1$ points in $PG(4,q)$. We show that if every 3-space meets $K$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic, then $K$ is a ruled cubic surface. Moreover, $K$ corresponds via the Bruck-Bose representation to a tangent Baer subplane of $PG(2,q^2)$. We use this to prove a characterisation in $PG(2,q^2)$ of a set of points $B$ as a tangent Baer subplane by looking at the intersections of $B$ with Baer-pencils.
In this article we look at a scroll of $PG(6,q)$ that uses a projectivity to rule a conic and a twisted cubic. We show this scroll is a ruled quintic surface $\mathcal V^5_2$, and study its geometric properties. The motivation in studying this scroll lies in its relationship with an $\mathbb F_q$-subplane of $PG(2,q^3)$ via the Bruck-Bose representation.
Let $H$ be a non-empty set of hyperplanes in $PG(4,q)$, $q$ even, such that every point of $PG(4,q)$ lies in either $0$, $\frac12q^3$ or $\frac12(q^3+q^2)$ hyperplanes of $ H$, and every plane of $PG(4,q)$ lies in $0$ or at least $\frac12q$ hyperplanes of $H$. Then $H$ is the set of all hyperplanes which meet a given non-singular quadric $Q(4,q)$ in a hyperbolic quadric.
In this article we construct a series of new infinite families of strongly regular graphs with the same parameters as the point-graphs of non-singular quadrics in PG(n,2).
In $\PG(2,q^3)$, let $π$ be a subplane of order $q$ that is exterior to $\li$. The exterior splash of $π$ is defined to be the set of $q^2+q+1$ points on $\li$ that lie on a line of $π$. 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 $CG(3,q)$, Sherk surfaces of size $q^2+q+1$, and $\GF(q)$-linear sets of rank 3 and size $q^2+q+1$. 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.
Let $π$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to $\ell_\infty$. The exterior splash of $π$ is the set of $q^2+q+1$ points on $\ell_\infty$ 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 of $PG(5,q)$. In this article we use the Bruck-Bose representation in $PG(6,q)$ to give a geometric characterisation of special conics of $π$ in terms of the covers of the exterior splash of $π$. We also investigate properties of order-$q$-subplanes\ with a common exterior splash, and study the intersection of two exterior splashes.
Let $π$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to $\ell_\infty$. Then the exterior splash of $π$ is the set of $q^2+q+1$ points on $\ell_\infty$ 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 $PG(5,q)$. In this article 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. In $PG(6,q)$, an exterior splash $\mathbb S$ has two sets of cover planes (which are hyper-reguli) and we show that each set has three unique transversals lines in the cubic extension $PG(6,q^3)$. These transversal lines are used to characterise the carriers of $\mathbb S$, and to characterise the sublines of $\mathbb S$.