F. De Clerck

An Infinite Class of Partial Geometries Associated with the Hyperbolic Quadric in PG(4n -1, 2)

We construct a class of partial geometries with parameters s e= 22n-1-1; t = 22n-1; α = 22n-2 associated with the hyperbolic quadric in PG(4n - 1, 2).

On Linear Representations of Near Hexagons

We discuss linear representations of near polygons in affine spaces. All linear representations of near hexagons in an affine space of orderq? 3 and dimension up to seven are classified. If the dimension of the affine space is at least eight, then the near hexagon necessarily contains a quad of typeT*2(O) and every such quad has a rosette of ovoids. We conjecture that there are no such examples.

The embedding of (0, α)-geometries in PG(n, q): part II

Abstract A (finite) (0,α)-geometry is an incidence structure S=(P, B, I) of points and lines with a symmetric incidence relation satisfying: (a) S is connected, (b) each line is incident with s+ 1 (s ≥ 1) points and two distinct lines are incident with at most one point, and (c) if a point x and a line L are not incident, then there are D or α (α ≥ 1) points which are collinear with x and incident with L. In the paper the (0,α)-geometries embeddable in PG (n,s) are considered. As a particular case all dual semi partial geometries embeddable in PG (n,s) are determined.

On linear representations of (a, b)-geometries

Abstract In [11] P. J. Cameron introduced partial quadrangles and raised the question of finding a characterization of partial quadrangles which have linear representations. An almost complete answer was given in [9]: the proof was a number-theoretic one. In this paper we discuss the question for a more general class of geometries, namely the (α, β)-geometries. We shall specialize to the case of (0,1)-geometries, and we shall give a geometric characterization of the partial quadrangle T*2 ( O ).

A geometric approach to Mathon maximal arcs

In 1969 Denniston [3] gave a construction of maximal arcs of degree d in Desarguesian projective planes of even order q, for all d dividing q. In 2002 Mathon [8] gave a construction method generalizing the one of Denniston. We will give a new geometric approach to these maximal arcs. This will allow us to count the number of isomorphism classes of Mathon maximal arcs of degree 8 in PG(2,2^h), h prime.

FLOCKS OF A QUADRATIC CONE IN PG(3, q), q ~< 8

If ℒ(O) is a quadratic cone in PG(3,q), with vertex x, then a flock of ℒ(O) is a partition of ℒ(O)-{x} into q disjoint conics. With such a flock there correspond a translation plane of order q2 and a generalized quadrangle of order (q2, q). Here we determine all flocks of ℒ(O) for q ≤ 8.

Subplane covered nets and semipartial geometries

Abstract Using the characterization theorems for (semi)partial geometries which satisfy the diagonal axiom, we prove that subplane covered nets or equivqlently (n − 1)-regulus nets are isomorphic to the dual of the geometry H qn+ 1 with point set, the set of points of a projective space ∑≅PG(n+1, q) which do not belong to a fixed subspace H ≅PG(n − 1, q) and with line set, the set of lines of ∑ skew to H . Moreover we discuss some combinatorial problems on subplane covered nets. Some of the results are known in the literature and have group theoretic proofs, our proofs however are geometrical.

A characterization of the sets of internal and external points of a conic

We provide a characterization of the sets of internal and external points of a nondegenerate conic in the plane PG(2,q), q odd, by means of their pattern of intersection with lines. In fact, we classify all sets of class [0,12(q-1),12(q+1),q] in PG(2,q), q odd.

On (0,α)-Geometries and Dual Semipartial Geometries Fully Embedded in an Affine Space

The (0,α)-geometries fully embedded in a projective space are up to a few open cases classified. For (0,α)-geometries fully embedded in an affine space AG(n,q), less is known. The most important model is the so-called linear representation Tn-1* (k) of a set k of type {0,1,α +1} with respect to lines in the hyperplane at infinity. We will give a characterization of this model. We also investigate the case where the (0,α)-geometry, fully embedded in AG(n,q), is the dual of a semipartial geometry.

A characterization of the partial geometry T*-2- (K)

In this paper we characterize the partial geometry T2* (K) embedded in AG(3, q) as a net-inducible partial geometry. This characterization is closely related to the characterization theorem of the generalized quadrangle T2*(O) in [8].