Related papers: A characterisation of Baer subplanes
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 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.
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…
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…
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…
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…
We consider the structure of the plane-line incidence matrix of the projective space $\mathrm{PG}(3,q)$ with respect to the orbits of planes and lines under the stabilizer group of the twisted cubic. Structures of submatrices with…
A smooth ruled surface in 4-space has only parabolic points or inflection points of real type. We show, by means of contact with transverse planes, that at a parabolic point, there exist two tangent directions determining two planes along…
We consider the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $\mathbb{F}_q$ into orbits of the group $PGL_2(q)$ of linear symmetries of the twisted cubic $C$. The problem has been solved in…
In this paper, we analyze the planar cubic Alternative curve to determine the conditions for convex, loops, cusps and inflection points. Thus cubic curve is represented by linear combination of three control points and basis function that…
Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that, the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the…
We consider the Andr\'e/Bruck-Bose representation of the projective plane $\mathrm{PG}(2,q^n)$ in $\mathrm{PG}(2n,q)$. We investigate the representation of $\mathbb{F}_{q^k}$-sublines and $\mathbb{F}_{q^k}$-subplanes of…
Let $\mathrm{PG}(3,q)$ be the projective space of dimension three over the finite field with $q$ elements. Consider a twisted cubic in $\mathrm{PG}(3,q)$. The structure of the point-plane incidence matrix in $\mathrm{PG}(3,q)$ with respect…
In this paper, we characterise ovoidal cones by their intersection numbers. We first show that a set of points of $\mathrm{PG}(4,q)$ which intersects planes in $1$, $q+1$ or $2q+1$ points is either an ovoidal cone or a parabolic quadric,…
We consider the action of the group $\mathrm{PGL}_4(K)$ on the smooth cubic surfaces of $\mathbb{P}^3_K$ ($K$ an algebraically closed field of characteristic zero). We classify, in an explicit way, all the smooth cubic surfaces with non…
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,…
In the projective space $\mathrm{PG}(3,q)$, we consider the orbits of lines under the stabilizer group of the twisted cubic. It is well known that the lines can be partitioned into classes every of which is a union of line orbits. All types…
We prove that a surface in real 3-space containing a line and a circle through each point is a quadric. We also give some particular results on the classification of surfaces containing several circles through each point.
We consider the structure of the point-line incidence matrix of the projective space $\mathrm{PG}(3,q)$ connected with orbits of points and lines under the stabilizer group of the twisted cubic. Structures of submatrices with incidences…