arXiv subjects

Wen-Ai Jackson

Publications and source records attributed to Wen-Ai Jackson.

At least 19 recordsLinked to original sources

arXiv2026

The planar projectivity of PG(2, $q^3$) of order 3 under field reduction

Let $ϕ$ 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 $ϕ$ : fixed points; points $P$ with $P, P^ϕ, P^{ϕ^{2}}$ distinct and collinear; and points $P$ with $P, P^ϕ, P^{ϕ^{2}}$ not collinear. Under field reduction, the collineation $ϕ$ corresponds to a projectivity $σ$ of $\operatorname{PG}(8, q)$ of order 3 . With respect to the field reduction and the orbits of $σ$, the points of $\mathrm{PG}(8, q)$ can be partitioned into six types. This article looks at the projectivity $σ$ 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.

math.CO↗

arXiv2026

The intersection of two linear sets of rank 3 in PG(1,q^3)

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.

math.CO↗

arXiv2026

Inherited arcs in André planes of even order

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.

math.CO↗

arXiv2026

Enumerating inherited conics in André planes of odd order

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.

math.CO↗

arXiv2025

The partition of $PG(2,q^3)$ arising from an order 3 planar collineation

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.

math.CO↗

arXiv2024

A Note on Fano Planes in Orthogonal Buekenhout-Metz Unitals of Even Order

An O'Nan configuration in a unital is a set of four lines forming a quadrilateral, where the six intersections of pairs of lines are points of the unital. In 2019 Feng and Li elegantly construct O'Nan configurations in orthogonal and Tits Buekenhout-Metz unitals. We extend their work by extending their construction to a Fano plane embedded in the orthogonal Buekenhout-Metz unital of even order. We deduce that there exist O'Nan configurations in orthogonal Buekenhout-Metz unitals different to those of Feng and Li, and make a conjecture about Fano planes embedded in orthogonal Buekenhout-Metz unitals.

math.CO↗

arXiv2024

Triple O'Nan Configurations in Buekenhout-Metz Unitals of Odd Order

An O'Nan configuration in a unital is a set of four lines forming a quadrilateral, where the six intersections of pairs of lines are points of the unital. In 2019 Feng and Li elegantly construct O'Nan configurations in Buekenhout-Metz unitals, in particular, for odd order unitals. We extend their work by showing the existence of Triple O'Nan configurations (a configuration containing three distinct O'Nan configurations) in these odd order Buekenhout-Metz unitals.

math.CO↗

arXiv2022

A characterisation of F_q-conics of PG(2,q^3)

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).

math.CO↗

arXiv2019

The Bose representation of PG(2,q^3) in PG(8,q)

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 $PG(8,q)$.

math.CO↗

arXiv2019

Specialness and the Bose representation

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 $PG(2,q^2)$.

math.CO↗

arXiv2019

Conics in Baer subplanes

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 $PG(2,q^2)$.

math.CO↗

arXiv2019

Characterising elliptic solids of $Q(4,q)$, $q$ even

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.

math.CO↗

arXiv2019

A characterisation of Baer subplanes

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.

math.CO↗

arXiv2019

Characterising hyperbolic hyperplanes of a non-singular quadric in $PG(4,q)$

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.

math.CO↗

arXiv2016

New families of strongly regular graphs

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).

math.CO↗

arXiv2015

Exterior splashes and linear sets of rank 3

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.

math.CO↗

arXiv2014

The exterior splash in $PG(6,q)$: Special conics

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.

math.CO↗

arXiv2014

The exterior splash in PG(6,q): Transversals

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 $\mathbb S$.

math.CO↗