We give a combinatorial characterization of the family of lines of P G(3, q) which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with the points and planes of PG(3,q).
We prove that, for $q$ odd, a set of $q+2$ points in the projective plane over the field with $q$ elements has at least $2q-c$ odd secants, where $c$ is a constant and an odd secant is a line incident with an odd number of points of the set.
In $PG(3,q^2)$, with $q$ odd, we determine the possible intersection sizes of a Hermitian surface $\mathcal{H}$ and an irreducible quadric $\mathcal{Q}$ having the same tangent plane $π$ at a common point $P\in{\mathcal Q}\cap{\mathcal H}$.
We show that a unital $U$ in $\mathrm{PG}(2,q^2)$ containing a point $P$, such that at least $q^2-ε$ of the secant lines through $P$ intersect $U$ in a Baer subline, is an ovoidal Buekenhout-Metz unital (where $ε\approx 2q$ for $q$ even and $ε\approx q^{3/2}/2$ for $q$ odd).
We constuct a family of hemisystems of the parabolic quadric $\mathcal{Q}(2d, q)$, for all ranks $d \ge 2$ and all odd prime powers $q$, that admit $Ω_3(q) \cong \mathrm{PSL}_2(q)$. This yields the first known construction for $d \ge 4$.
We use the representation $T_2(O)$ for $\q(4,q)$ to show that maximal partial ovoids of $\q(4,q)$ of size $q^2-1$, $q=p^h$, $p$ odd prime, $h > 1$, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for $q$ prime.
We prove that the parameter $x$ of a tight set $\mathcal{T}$ of a hyperbolic quadric $\mathsf{Q}^+(2n+1,q)$ of an odd rank $n+1$ satisfies ${x\choose 2}+w(w-x)\equiv 0\mod q+1$, where $w$ is the number of points of $\mathcal{T}$ in any generator of $\mathsf{Q}^+(2n+1,q)$. As this modular equation should have an integer solution in $w$ if such a $\mathcal{T}$ exists, this condition rules out roughly at least one half of all possible parameters $x$. It generalizes a previous result by the author and K. Metsch shown for tight sets of a hyperbolic quadric $\mathsf{Q}^+(5,q)$ (also known as Cameron…
A $pseudo$-$oval$ of a finite projective space over a finite field of odd order $q$ is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order $(q^n,q^n)$ and a Laguerre plane of order $q^n$ (for some $n$). In setting out a programme to construct new generalised quadrangles, Shult and Thas asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric ${Q}^-(5,q)$, non-equivalent to the $classical$ $example$, a so-called $pseudo$-$conic$. To date, every known pseudo-oval of lines of ${Q}^-(5,q)$ is proje…
Let $\mathcal{O}$ be a conic in the classical projective plane $PG(2,q)$, where $q$ is an odd prime power. With respect to $\mathcal{O}$, the lines of $PG(2,q)$ are classified as passant, tangent, and secant lines, and the points of $PG(2,q)$ are classified as internal, absolute and external points. The incidence matrices between the secant/passant lines and the external/internal points were used in \cite{keith1} to produce several classes of structured low-density parity-check binary codes. In particular, the authors of \cite{keith1} gave conjectured dimension formula for the binary code $\ma…
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 point to C, the set of all nontangent lines to C through a point in C, and, for square q, the set of all nontangent lines to C belonging to a Baer subplane with at least 5 common points with C. This classification theorem is the analogous of a classical result by Segre and Korchmaros characterizing the pencil of lines through an internal point to C as the u…
It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of combinatorial Grassmannian of type $G_2(7)$, $\mathcal{V}(G_2(7))$. The lines of the ambient symplectic polar space are those lines of $\mathcal{V}(G_2(7))$ whose cores feature an odd number of points of $G_2(7)$. After introducing basic properties of three different types of points and six distinct types of lines of $\mathcal{V}(G_2(7))$, we explicitly show the combinatorial Grassmannian composition of the magic Veldkamp line; we first give representatives of points and lines of its core …
In this paper, we describe a new infinite family of $\frac{q^{2}-1}{2}$-tight sets in the hyperbolic quadrics $\mathcal{Q}^{+}(5,q)$, for $q \equiv 5 \mbox{ or } 9 \bmod{12}$. Under the Klein correspondence, these correspond to Cameron--Liebler line classes of ${\rm PG}(3,q)$ having parameter $\frac{q^{2}-1}{2}$. This is the second known infinite family of nontrivial Cameron--Liebler line classes, the first family having been described by Bruen and Drudge with parameter $\frac{q^{2}+1}{2}$ in ${\rm PG}(3,q)$ for all odd $q$. The study of Cameron--Liebler line classes is closely related to the …
The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $\mathbb{C}$ and $\mathbb{R}$ in 1906--1907. The analogous problem for finite fields $\mathbb{F}_q$ with $q$ odd was solved by Dickson in 1908. In 1914, Wilson attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson's classification of nets of rank one, namely those cont…
