Related papers: Classification of flocks of the quadratic cone in …

This is the first in a series of articles devoted to providing a foundation for a theory of flocks of arbitrary cones in PG(3,q). The desire to have such a theory stems from a need to better understand the very significant and applicable…

The concept of a flock of a quadratic cone is generalized to arbitrary cones. Flocks whose planes contain a common point are called star flocks. Star flocks can be described in terms of their coordinate functions. If the cone is "big…

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…

Ovoids of the non-degenerate quadric Q(4,q) of PG(4,q) have been studied since the end of the '80s. They are rare objects and, beside the classical example given by an elliptic quadric, only three classes are known for q odd, one class for…

Ovoids in $\PG(3, q)$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families are known. The first is the elliptic quadratics and the second is the Tits ovoids. In this…

In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order $(5^2,5)$ and give numerous further examples of hemisystems in all the known flock generalized…

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…

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.

Brown et al. provide a representation of a spread of the Tits quadrangle $T_2(\mathcal O)$, $\mathcal O$ an oval of $\mathrm PG(2,q)$, $q$ even, in terms of a certain family of $q$ ovals of $\mathrm PG(2,q)$. By combining this…

In this paper we study combinatorial invariants of the equivalence classes of pencils of cubics on $\mathrm{PG}(1,q)$, for $q$ odd and $q$ not divisible by 3. These equivalence classes are considered as orbits of lines in…

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 paper, we revisit the theory of perfect unary forms over real quadratic fields. Specifically, we deduce an infinite family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ when $d=2$ or $3$ mod $4$, such that there are three classes…

We consider the orbits of the group $G=PGL_2(q)$ on the points, lines and planes of the projective space $PG(3,q)$ over a finite field $\mathbb F_q$ of characteristic different from $2$ and $3$. The points of $PG(3,q)$ can be identified…

Ovoids of the Klein quadric $Q^+(5,q)$ of $\mathrm{PG}(5,q)$ have been studied in the last 40 year, also because of their connection with spreads of $\mathrm{PG}(3,q)$ and hence translation planes. Beside the classical example given by a…

We consider the cyclic presentation of $PG(3,q)$ whose points are in the finite field $\mathbb{F}_{q^4}$ and describe the known ovoids therein. We revisit the set $\mathcal{O}$, consisting of $(q^2+1)$-th roots of unity in…

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…

A quadratic line complex is a three-parameter family of lines in projective space P^3 specified by a single quadratic relation in the Plucker coordinates. Fixing a point p in P^3 and taking all lines of the complex passing through p we…

In the projective space $\mathrm{PG}(3,q)$, we consider the orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of $\mathrm{PG}(3,q)$ are partitioned into classes, each of which is a union of line…

Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of $n$ squares and two cone points with angle $4 \pi$ each, we set up and parametrize…

This paper completes the classification of nets of conics containing at least one double line in $\mathrm{PG}(2,q)$ for $q$ even. This classification contributes to the classification of partially symmetric tensors in $\mathbb{F}_q^3…