Conical flocks,partial flocks,derivation, and generalized quadrangles

L. Bader, G. Lunardon and J. A. Thas have shown that a flock ₀ of a quadratic cone in PG(3, q), q odd, determines a set ={⁰,₁,...,_q} of q+1 flocks. Each _j , 1jq, is said to be derived from ₀. We show that, by derivation, the flocks with q=3 ^e arising from the Ganley planes yield an inequivalent flock for q27. Further, we prove that the Fisher flocks (q odd, q5) are the unique nonlinear flocks for which (q–1)/2 planes of the flock contain a common line. This result is used to show that each of the flocks derived from a Fisher flock is again a Fisher flock. Finally, we prove that any set of q–1 pairwise disjoint nonsingular conics of a cone can be extended to a flock. All these results have implications for the theory of translation planes.     

Spreads and ovoids in finite generalized quadrangles

This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s ²), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3 ^e ),e3, is constructed. Then, by the duality betweenQ(4, 3 ^e ) and the classical generalized quadrangleW (3 ^e ), we get line spreads of PG(3, 3 ^e ) and hence translation planes of order 3^2e . These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q ²,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q ²),q even, having a subquadrangleS isomorphic toQ(4,q) and if inS each ovoid consisting of all points collinear with a given pointx ofS\S is an elliptic quadric, thenS is isomorphic toQ(5,q).     

Translation generalized quadrangles

The author gratefully acknowledges the financial support which he received from the Austrian Fonds zur Förderung der wissenschaftlichen Forschung during parts of the work on this paper.     

Extended generalized quadrangles

Extended generalized quadrangles (roughly, connected structures whose every residue is a generalized quadrangle) are studied in some detail, especially those which are uniform or strongly uniform. Much basic structure theory is developed, many examples are given, and something approaching characterization is given for many types.Dedicated to Professor Jacques Tits for his sixtieth birthday     

Extending generalized quadrangles

Extended generalized quadrangles (EGQ) are the geometries associated with the Buekenhout diagram , where is the diagram for generalized quadrangles. In this paper we survey the two cases where an (EGQ) is either a 2-design or a locally polar space of polar rank 2.     

Nets and generalized quadrangles

There is a new method of constructing generalized quadrangles (GQs) given by S. Löwe, which is based on covering of nets; all GQs with a regular point can be represented in this way. Here we give a method of constructing GQs with a regular point using the so-called content functions on nets. In the last part of the paper we lay the foundations for a research project aiming to use the more general notion of content to classify GQs and maybe to construct new ones.Both authors acknowledge the financial support by CRUI and DAAD in the frame of Programma Vigoni, which made this work possible.     

Finite Moufang generalized quadrangles

This paper gives a review of several equivalent Moufang conditions and the corresponding geometric classification in the finite case. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 22, Algebra and Geometry, 2004.     

Triangular extended generalized quadrangles

In this paper, we continue the study from [2] of what are now called triangular extended generalized quadrangles. In particular, we determine all parameter sets such that the point graph is strongly regular with intersection number =2(t+1).     

Classical Klingenberg generalized quadrangles

Research supported by I.W.O.N.L. grant no. 840037.     

Characterizations of generalized quadrangles by generalized homologies

Let S = (P, B, I) be a generalized quadrangle of order (s, t). For x, y P, x y, let (x, y) be the group of all collineations of S fixing x and y linewise. If z {x, y}, then the set of all points incident with the line xz (resp. yz) is denoted by (resp. ). The generalized quadrangle S = (P, B, I) is said to be (x, y)-transitive, x y, if (x, y) is transitive on each set and . If S = (P, B, I) is a generalized quadrangle of order (s, t), s > 1 and t > 1, which is (x, y)-transitive for all x, y P with x y, then it is proved that we have one of the following: (i) S W(s), (ii) S Q(4, s), (iii) S H(4, s), (iv) S Q(5, s), (v) s = t² and all points are regular.     

Semi-regular graph automorphisms and generalized quadrangles

We use topological techniques to verify the existence of non-trivial semi-regular automorphisms of certain graphs, and use this to obtain a characterization of certain families of Moufang quadrangles.     

Notes on elation generalized quadrangles

Let be a finite generalized quadrangle of order (s,t),s,t>1. An “elation about a point p” of is an automorphism fixing p linewise and fixing no point which is not collinear with p. An elation that generates a cyclic group of elations is called a “standard elation”. One of the problems already considered in Payne and Thas (Finite Generalized Quadrangles (1984)) is to determine just when the set of elations about the point (∞) is a group. The purpose of this paper is to provide an example where this is not the case, and then to show that for a flock generalized quadrangle the usual group of elations about (∞) is the complete set of standard elations about (∞).     

On embeddings of Ahrens-Szekeres generalized quadrangles

We construct a series of embeddings of the ASGQ's into affine 3-space, parameterized by the points of an affine plane. Points and lines of the 3-space, together with the images of all embeddings, form a diagram geometry of rank 3.Herrn Professor Helmut Karzel zum 70. Geburtstag gewidmet     

Affine ovoids and extensions of generalized quadrangles

A set Δ of vertices of a generalized quadrangle of order (s, t) is said to be a hyperoval if any line intersects Δ in either 0, or 2 points. A hyperoval Δ is called an affine ovoid if |Δ|=2st. It is well known that μ-subgraphs in triangular extensions of generalized quadrangles are hyperovals. In the present paper we prove that ifS is a triangular extension forGQ(s, t) with totally regular point graph Γ such that μ=2st, thens is even, Γ is an τ-antipodal graph of diameter 3 with τ=1+s/2, and eithers=2, ort=s+2. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 266–271, August, 2000.     

Flocks of quadratic cones,generalized quadrangles and translation planes

Geometriae Dedicata -     

An essay on self-dual generalized quadrangles

We collect some known facts about a self-dual generalized quadrangle (GQ) and consider especially the number of absolute points of a duality. The only known finite self-dual GQs are the ${T_2(\mathcal O)}$ constructed by J. Tits where ${\mathcal O}$ is a translation oval in a finite desarguesian projective plane of even order. We review a construction of these GQs in enough detail to study the sets of absolute points of certain dualities, but for more details about these GQs see the monograph Finite Generalized Quadrangles (Europ. Math. Soc., Zurich 2009). After some generalities about the incidence matrix of a finite (GQ) we return to the study of self-dual examples.     

Hemisystems of small flock generalized quadrangles

In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order (5², 5) and give numerous further examples of hemisystems in all the known flock generalized quadrangles of order (s ², s) for s ≤ 11. By analysing the computational data, we identify two possible new infinite families of hemisystems in the classical generalized quadrangle H(3, s ²).