Intriguing sets in partial quadrangles

The point‐line geometry known as a partial quadrangle (introduced by Cameron in 1975) has the property that for every point/line non‐incident pair (P, ?), there is at most one line through P concurrent with ?. So in particular, the well‐studied objects known as generalized quadrangles are each partial quadrangles. An intriguing set of a generalized quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalized quadrangles by Bamberg, Law and Penttila to partial quadrangles, which gives insight into the structure of hemisystems and other intriguing sets of generalized quadrangles. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:217‐245, 2011     

Local Sharply Transitive Actions on Finite Generalized Quadrangles

We classify the finite generalized quadrangles containing a line L such that some group of collineations acts sharply transitively on the ordered pentagons which start with two points of L. This can be seen as a generalization of a result of Thas and the second author [22] classifying all finite generalized quadrangles admitting a collineation group that acts transitively on all ordered pentagons, although the restriction to sharp transitivity is essential in our arguments. However, the conclusion is exactly the same family of classical generalized quadrangles (the orthogonal quadrangles and their duals). Our main result thus provides a local group theoretic characterization of these classical quadrangles.     

Generalized quadrangles admitting a sharply transitive Heisenberg group

All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.      

Tight sets,weighted <Emphasis Type="Italic">m</Emphasis>-covers,weighted <Emphasis Type="Italic">m</Emphasis>-ovoids,and minihypers

Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342–349, 2008; Govaerts and Storme 4:279–286, 2004; Govaerts et al. 28:659–672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q⁻(5,3), to 56-caps contained in this elliptic quadric Q⁻(5,3).      

A generalized quadrangle of order (s, t) with center of transitivity is an elation quadrangle if s ≤ t

We show that a generalized quadrangle of order (s, t) with a center of transitivity is an elation generalized quadrangle if s ≤ t. In order to obtain this result, we generalize Frohardt’s result on Kantor’s conjecture from elation quadrangles to the more general case of quadrangles with a center of transitivity.      

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).     

Determination of generalized quadrangles with distinct elation points

In this paper, we classify the finite generalized quadrangles of order (s,t), s,t > 1, which have a line L of elation points, with the additional property that there is a line M not meeting L for which {L, M} is regular. This is a first fundamental step towards the classification of those generalized quadrangles having a line of elation points. Mathematics Subject Classification (2000): 51E12, 51E20, 20B25, 20E42     

Affine Generalized Quadrangles – An Axiomatization

The complement of a geometric hyperplane of a generalized quadrangle is called an affine generalized quadrangle. Since a geometric hyperplane of a generalized quadrangle is either an ovoid or the perp of a point or a subquadrangle, there are three quite different classes of affine generalized quadrangles. The article proposes seven axioms (AQ1)–(AQ7) characterizing affine generalized quadrangles as point-line geometries. Certain subsets of the seven Axioms together with certain conditions distinguish what kind of hyperplane complement is realized. By just (AQ1)–(AQ6), finite affine generalized quadrangles are characterized completely.     

Slanted symplectic quadrangles

By slanting symplectic quadrangles W(F) over fieldsF, we obtain very simple examples of non-classical generalized quadrangles. We determine the collineation groups of these slanted quadrangles and their groups of projectivities. No slanted quadrangle is a topological quadrangle.     

Multiple intersections of diagonals of regular polygons,and related topics

Our main concern is to investigate geometrically all sets of three concurrent chords of regular polygons or, equivalently, all adventitious quadrangles (that is, all quadrangles such that the angle between every pair of the six sides is an integral multiple of /n radians). Most of our results are stated without proof. The proofs are elementary, often consisting of straightforward verification; to include them would make the paper much longer and less readable.     

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     

An essay on skew translation generalized quadrangles

Astract We study a class of generalized quadrangles satisfying a local Moufang condition that is more restrictive than the usual concept of skew translation GQ. A new infinite family is constructed whose point-line duals are translation generalized quadrangles with very small kernel. They have parameters (q ², q), q=3 ^r , for r>2.This paper was written while the author was supported by the CNR as a visiting professor at the University of Rome La Sapienza.     

SPG-reguli, SPG-systems, BLT-sets and sets with the BLT-property

This paper is a survey on SPG-reguli, SPG-systems, BLT-sets and sets with the BLT-property. It is shown how from these sets generalized quadrangles, partial geometries and semi-partial geometries can be constructed. Many examples are given and open problems are stated. There are also some new results.     

Lie Tori of Type BC 2 and Structurable Quasitori

Structurable quasitori are introduced and shown to give all Lie tori of type BC ₂ via the Kantor construction. Structurable quasitori, including Jordan quasitori, are classified. The methods include a use of generalized quadrangles of order (2,t).     

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).     

Generalized quadrangles from a local point of view

In this lecture, I will survey several recent results in the local theory of generalized quadrangles. Starting with a short introduction to the global automorphism theory, I will motivate as such the local viewpoint, and overview some of the most important local properties which are investigated nowadays. Recent results on skew translation quadrangles and forms will be described, including a solution of a question of Payne which generalizes work of Havas et?al. (Finite geometries, groups, and computation, 2006; Adv Geom 26:389?C396, 2006), and then I will mention parts of a classification of skew translation quadrangles which is being prepared by the author. Finally, I will consider conditions which are both global and local.     

Blocking sets of Hermitian generalized quadrangles

Some infinite families of minimal blocking sets on Hermitian generalized quadrangles are constructed.     

Difference of two sets and estimation of clarke generalized Jacobian via quasidifferential

The notion of difference for two convex compact sets inR ⁿ , proposed by Rubinovet al, is generalized toR ^m×n . A formula of the difference for the two sets, which are convex hulls of a finite number of points, is developed. In the light of this difference, the relation between Clarke generalized Jacobian and quasidifferential, in the sense of Demyanov and Rubinov, for a nonsnooth function, is established. Based on the relation, the method of estimating Clarke generalized Jacobian via quasidifferential for a certain class of functions, is presented.     

Symmetry in Generalized Quadrangles

In this paper, we describe some aspects of a Lenz(-Barlotti)-type classification of finite generalized quadrangles, which is being prepared by the author. Some new points of view are given. We also prove that each span-symmetric generalized quadrangle of order s > 1 with s even is isomorphic to $ \mathcal{Q} $ (4, s), without using the canonical connection (obtained by S. E. Payne in [15] between groups of order s ³ ? s with a 4-gonal basis and span-symmetric generalized quadrangle of order s. (The latter result was obtained for general s independently by W. M. Kantor in [10], and the author in [30] Finally, we obtain a classification program for all finite translation generalized quadrangles, which is suggested by the main results of [27], [30], [32], [35], [38] and [37].     

Antidesigns and regularity of partial spreads in dual polar graphs

We give several examples of designs and antidesigns in classical finite polar spaces. These types of subsets of maximal totally isotropic subspaces generalize the dualization of the concepts of m ‐ovoids and tight sets of points in generalized quadrangles. We also consider regularity of partial spreads and spreads. The techniques that we apply were developed by Delsarte. In some polar spaces of small rank, some of these subsets turn out to be completely regular codes. © 2010 Wiley Periodicals, Inc. J Combin Designs 19: 202‐216, 2011