H. Van Maldeghem

Translation Ovoids of Generalized Quadrangles and Hexagnos

We define the notion of a translation ovoid in the classical generalized quadrangles and hexagons of order q, and we enumerate all known examples; translation spreads are defined dually. A modification of the known ovoids in the generalized hexagon H(q), q=32h+1, yields new ovoids of that hexagon. Dualizing and projecting along reguli, we obtain an alternative construction of the Roman ovoids due to Thas and Payne. Also, we construct a new translation spread in H(q) for any ≡ 1 mod 3, q odd, with the property that any projection along reguli yields the classical ovoid in the generalized quadrangle Q(4,q). Finally, we prove that for q odd, the new example is the only non-Hermitian translation spread in H(q) with the property that any projection along reguli yields the classical ovoid in Q(4,q).

Finite distance-transitive generalized polygons

Using the classification of the finite simple groups, we classify all finite generalized polygons having an automorphism group acting distance-transitively on the set of points. This proves an old conjecture of J. Tits saying that every group with an irreducible rank 2 BN-pair arises from a group of Lie type.

Non-classical triangle buildings

In [7], J. Tits classifies the affine buildings of rank greater or equal to 4. That leaves the question of the rank 3 buildings. This paper shows that the class of all triangle buildings is as wild as the class of all projective planes.

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 ).

Some new two-character sets in PG(5, q2) and a distance-2 ovoid in the generalized hexagon H(4 )

In this paper, we construct a new infinite class of two-character sets in PG(5,q^2) and determine their automorphism groups. From this construction arise new infinite classes of two-weight codes and strongly regular graphs, and a new distance-2 ovoid of the split Cayley hexagon of order 4.

Cyclic Arcs in PG (2, q )

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q2 − q + l)-arcs in PG(2, q2), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q2 − q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.

Embeddings of small generalized polygons

In this paper we consider some finite generalized polygons, defined over a field with characteristic 2, which admit an embedding in a projective or affine space over a field with characteristic unequal to 2. In particular, we classify the (lax) embeddings of the unique generalized quadrangle H(3,4) of order (4,2). We also classify all (lax) embeddings of both the split Cayley hexagon H(2) and its dual H(2)^d^u^a^l in 13-dimensional projective space PG(13,K), for any skew field K. We apply our results to classify the homogeneous embeddings of these small generalized hexagons, and to classify all homogeneous lax embeddings in real spaces of them. Also, we classify all homogeneous embeddings of generalized quadrangles of order (2,2), (4,2) and (2,4).

Classification of finite Veronesean caps

We show that all Veronesean caps in finite projective spaces, as defined by Mazzocca and Melone (Discrete Math. 48 (1984) 243), are projections of quadric Veroneseans. In fact we prove a slightly stronger result by weakening one of the conditions of Mazzocca and Melone.

Flat Lax and Weak Lax Embeddings of Finite Generalized Hexagons

In this paper we study laxly embedded generalized hexagons in finite projective spaces (a generalized hexagon is laxly embedded inPG(d,q) if it is a spanning subgeometry of the natural point-line geometry associated toPG(d,q)), satisfying the following additional assumption: for any pointxof the hexagon, the set of points collinear in the hexagon withxis contained in some plane ofPG(d,q). In particular, we show thatd?7, and ifd=7, we completely classify all such embeddings. A classification is also carried out ford=5, 6 under some additional hypotheses. Finally, laxly embedded generalized hexagons satisfying other additional assumptions are considered, and classifications are also obtained.

Generalized quadrangles weakly embedded in finite projective space

Abstract We show that every weak embedding of any finite thick generalized quadrangle of order (s,t) in a projective space PG(d,q), q a prime power, is a full embedding in some subspace PG(d,s), where GF(s) is a subfield of GF(q), except in some well-known cases where we classify these exceptions. This generalizes a result of Lefevre-Percsy (1981, European J. Combin. 2, 249–255). who considered the case d=3.