Related papers: Characterising pointsets in PG(4,q) that correspon…

This article studies conics and subconics of $PG(2,q^2)$ and their representation in the Andr\'e/Bruck-Bose setting in $PG(4,q)$. In particular, we investigate their relationship with the transversal lines of the regular spread. The main…

In this article, a combinatorial characterization of the family of planes of $\PG(3,q)$ which meet a hyperbolic quadric in an irreducible conic, using their intersection properties with the points and lines of $\PG(3,q)$, is given.

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…

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, $K$…

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 $\frac12q$…

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…

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…

In this article, a combinatorial characterization of the family of parabolic hyperplanes of a hyperbolic (respectively, elliptic) quadric of PG(2n + 1, q), using their intersection properties with the points and subspaces of codimension 2,…

We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) when q is even. This characterisation is based on the number of elements of these sets through points and codimension…

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of ${\rm PG}(2,q)$ remains an arc in the Hall plane obtained by derivation. Some…

A set without tangents in $\PG(2,q)$ is a set of points S such that no line meets S in exactly one point. An exterior set of a conic $\mathcal{C}$ is a set of points $\E$ such that all secant lines of $\E$ are external lines of…

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…

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…

A unital in PG(2,q^2) is a set U of q^3+1 points such that each line meets U in 1 or q+1 points. The well known example is the classical unital consisting of all absolute points of a non-degenerate unitary polarity of PG(2,q^2). Unitals…

Let $\pi$ be an order-$q$-subplane of $PG(2,q^3)$ that is exterior to $\ell_\infty$. The exterior splash of $\pi$ is the set of $q^2+q+1$ points on $\ell_\infty$ that lie on an extended line of $\pi$. Exterior splashes are projectively…

We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent…

Let $\A$ be the incidence matrix of lines and points of the classical projective plane $PG(2,q)$ with $q$ odd. With respect to a conic in $PG(2,q)$, the matrix $\A$ is partitioned into 9 submatrices. The rank of each of these submatices…

We present a new construction of non-classical unitals from a classical unital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model $\Pi$ of $PG(2,q^2)$ with the following three…

In this paper, we investigate the Andr\'e/Bruck-Bose representation of certain $\mathbb{F}_q$-linear sets contained in a line of $\text{PG}(2,q^t)$. We show that scattered $\mathbb{F}_q$-linear sets of rank $3$ in $\text{PG}(1,q^3)$…

All sets of lines providing a partition of the set of internal points to a conic C in PG(2,q), q odd, are determined. There exist only three such linesets up to projectivities, namely the set of all nontangent lines to C through an external…