相关论文: A higgledy-piggledy set of planes based on the ABB…

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…

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…

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…

This work focuses on higgledy-piggledy sets of $k$-subspaces in $\text{PG}(N,q)$, i.e. sets of projective subspaces that are 'well-spread-out'. More precisely, the set of intersection points of these $k$-subspaces with any $(N-k)$-subspace…

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: 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…

We study the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $GF(q)$ into orbits of the group $G=PGL(2,q)$ of linear symmetries of the twisted cubic $C$. A generic line neither intersects $C$ nor…

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 paper we study a family of scattered $\F_q$--linear sets of rank $tn$ of the projective space $PG(2n-1,q^t)$ ($n \geq 1$, $t\geq 3$), called of {\it pseudoregulus type}, generalizing results contained in [G. Marino, O. Polverino, R.…

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 [2] and [19] are presented the first two families of maximum scattered $\mathbb{F}_q$-linear sets of the projective line $\mathrm{PG}(1,q^n)$. More recently in [23] and in [5], new examples of maximum scattered $\mathbb{F}_q$-subspaces…

There are two known families of maximum scattered $\mathbb{F}_q$-linear sets in $PG(1,q^t)$: the linear sets of pseudoregulus type and for $t\geq 4$ the scattered linear sets found by Lunardon and Polverino. For $t=4$ we show that these are…

In this article, we investigate collections of `well-spread-out' projective (and linear) subspaces. Projective $k$-subspaces in $\mathsf{PG}(d,\mathbb{F})$ are in `higgledy-piggledy arrangement' if they meet each projective subspace of…

We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then…

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…

In this paper, we study translation hyperovals in PG$(2,q^k)$. The main result of this paper characterises the point sets defined by translation hyperovals in the Andr\'e/Bruck-Bose representation. We show that the affine point sets of…

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…

We examine sets of lines in PG(d,F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least 1.5d lines if the field F has more than 1.5d elements, and at least 2d-1 lines if the field F is…

Let $\mathrm{PG}(1,E)$ be the projective line over the endomorphism ring $E=End_q({\mathbb F}_{q^t})$ of the $\mathbb F_q$-vector space ${\mathbb F}_{q^t}$. As is well known there is a bijection $\Psi:\mathrm{PG}(1,E)\rightarrow{\cal…

In this paper, we study the weight distributions of $\mathbb{F}_q$-linear sets in $\mathrm{PG}(1,q^5)$. Our main theorem proves that a linear set $S$ of rank $5$, which is not scattered has the following weight distribution for its points…