Related papers: Hyper-reguli in PG(5,q)

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…

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…

Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and…

Regular maps on linear fractional groups $PSL(2,q)$ and $PGL(2,q$) have been studied for many years and the theory is well-developed, including generating sets for the asscoiated groups. This paper studies the properties of self-duality,…

We construct an infinite family of intriguing sets that are not tight in the Grassmann Graph of planes of PG$(n,q)$, $n\ge 5$ odd, and show that the members of the family are the smallest possible examples if $n\ge 9$ or $q\ge 25$.

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…

Given an odd representation of the absolute Galois group of Q onto PGL(2,3) and a positive integer N, there exists a twisted modular curve defined over Q whose rational points classify the quadratic Q-curves of degree N realizing the…

We compute the fundamental groups of five maximizing sextics with double singular points only; in four cases, the groups are as expected. The approach used would apply to other sextics as well, given their equations.

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…

Let X be a non-singular projective hypersurface of degree 4, which is defined over the rational numbers. Assume that X has dimension 39 or more, and that X contains a real point and p-adic points for every prime p. Then X is shown to…

This paper studies {\em strong blocking sets} in the $N$-dimensional finite projective space $\mathrm{PG}(N,q)$. We first show that certain unions of blocking sets cannot form strong blocking sets, which leads to a new lower bound on the…

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.…

Minimal codes are being intensively studied in last years. $[n,k]_{q}$-minimal linear codes are in bijection with strong blocking sets of size $n$ in $PG(k-1,q)$ and a lower bound for the size of strong blocking sets is given by…

The maximum scattered linear sets in $PG(1,q^n)$ have been completely classified for $n \le 4$ by Csajb\'ok-Zanella and Lavrauw-Van de Voorde. Here a wide class of linear sets in $PG(1,q^5)$ is studied which depends on two parameters.…

We show that each set of $n\ge 2$ points in the plane in general position has a straight-line matching with at least $(5n+1)/27$ edges whose segments form a connected set, and such a matching can be computed in $O(n \log n)$ time. As an…

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…

A pseudo-hyperoval of a projective space $\PG(3n-1,q)$, $q$ even, is a set of $q^n+2$ subspaces of dimension $n-1$ such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabiliser is…

This is a summary of the work in the author's recent paper with this title written with Jerome Gauntlett, George Papadopoulos and Paul Townsend, hep-th/9702202. We showed how to construct hyper-K\"ahler 8-metrics in terms of arrangements 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,…