Spreads,ovoids andS 5

We consider translation planes of orderq ² (whereq andq ² - 1 are coprime to 30) such thatS ₅ acts on the line at infinity. It turns out that the Klein correspondence is in particular useful for the investigation of these planes. Representations of the planes, automorphisms and examples of low order are studied in detail. In view of a problem of Ostrom (Math. Z. 156 (1977), 59–71), series of translation planes are constructed with the following property: the translation complement is nonsolvable and has an order coprime to the characteristic of the plane.     

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.     

A note on isoparametric generalized quadrangles

We generalize a result of Kramer, see [7, 10.7 and 10.10], on generalized quadrangles associated with isoparametric hypersurfaces of Clifford type to Tits buildings of type C₂ derived from arbitrary isoparametric hypersurfaces with four distinct principal curvatures in spheres: two distinct points p and q of a generalized quadrangle associated with an isoparametric hypersurface in the unit sphere of a Euclidean vector space can be joined by a line K if and only if (p − q)/||p − q|| is a line. This line is orthogonal to K. Dually, two distinct lines L and K intersect if and only if (L − K)/||L − K|| is point. Received: 14 October 2005     

Two-transitive orbits in finite projective planes

The problem of classifying finite projective planes of order n with an automorphism group G and a point orbit on which G acts two-transitively is investigated in considerable detail, under the assumption that has length at last n. Combining old and new results a rather satisfying classification is obtained, even though some cases for orbit lengths n and n + 1 remain unsolved.     

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.     

A completion problem for finite affine planes

A partial affine plane (PAP) of ordern is ann ²-setS of points together with a collection ofn-subsets ofS called lines such that any two lines meet in at most one point. We obtain conditions under which a PAP with nearlyn ²+n lines can be completed to an affine plane by adding lines. In particular, we make use of Bruck’s completion condition for nets to show that certain PAP’s with at leastn ²+n−√n can be completed and that forn≠3 any PAP withn ²+n−2 lines can be completed.     

Sharp homogeneity in some generalized polygons

We show that, if a collineation group G of a generalized (2n + 1)-gon $\Gamma$ has the property that every symmetry of any apartment extends uniquely to a collineation in G, then $\Gamma$ is the unique projective plane with 3 points per line (the Fano plane) and G is its full collineation group. A similar result holds if one substitutes apartment with path of length 2k 2n + 2.Received: 19 June 2002     

Spreads,flocks, and generalized quadrangles

A spread of PG(3,q), q an odd prime, recently constructed by R. Baker and G. Ebert, when generalized for q an odd prime power is isomorphic to a spread derived by J. A. Thas from a flock of a quadratic cone discovered by J. C. Fisher. The associated generalized quadrangle has an unusual colllneation.     

Planar functions over finite fields

Letp>2 be a prime. A functionf: GF(p)GF(p) is planar if for everyaGF(p) ^*, the functionf(x+a–f(x) is a permutation ofGF(p). Our main result is that every planar function is a quadratic polynomial. As a consequence we derive the following characterization of desarguesian planes of prime order. IfP is a protective plane of prime orderp admitting a collineation group of orderp ², thenP is the Galois planePG(2,p). The study of such collineation groups and planar functions was initiated by Dembowski and Ostrom [3] and our results are generalizations of some results of Johnson [8].We have recently learned that results equivalent to ours have simultaneously been obtained by Y. Hiramine and D. Gluck.     

Generalized quadrangles with an ovoid that is translation with respect to opposite flags

The article [6] contains the result that if a finite generalized quadrangle of order s has an ovoid that is translation with respect to two opposite flags, but not with respect to any two non-opposite flags, then is self-polar and is the set of absolute points of a polarity. In particular, if is the classical generalized quadrangle Q(4, q) then is a Suzuki-Tits ovoid. In this article, we remove the need to assume that is Q(4, q) in order to conclude that is a Suzuki-Tits ovoid by showing that the initial assumptions in fact imply that is Q(4, q). At the same time, we also relax the requirement that have order s.Received: 14 May 2004     

On homomorphisms between generalized polygons

We consider homomorphisms between abstract, topological, and smooth generalized polygons. It is shown that a continuous homomorphism is either injective or locally constant. A continuous homomorphism between smooth generalized polygons is always a smooth embedding. We apply this result to isoparametric submanifolds.Dedicated to Prof. Dr. H. R. Salzmann on the occasion of his 65th anniversary     

Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

Maximal partial ovoids and maximal partial spreads of the hermitian generalized quadrangles H(3,q²) and H(4,q²) are studied in great detail. We present improved lower bounds on the size of maximal partial ovoids and maximal partial spreads in the hermitian quadrangle H(4,q²). We also construct in H(3,q²), q=2^2h+1, h≥ 1, maximal partial spreads of size smaller than the size q²+1 presently known. As a final result, we present a discrete spectrum result for the deficiencies of maximal partial spreads of H(4,q²) of small positive deficiency δ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 101–116, 2008     

The correlations of finite Desarguesian planes, Part I: Generalities

The polarities of Desarguesian planes have long been known. This author has undertaken to classify the correlations of finite Desarguesian planes in general. In [6] we have presented all the correlations with identity companion automorphism which are not polarities, of these planes. In this sequence of papers, we classify the correlations of planes of order $ p^{2^{i}(2n+1)}, n \neq 0 $, with companion automorphism ( $p^{2^{i}t}$ ), p an odd prime, $ t \neq 0 $. This represents a complete classification of the correlations of planes of odd nonsquare order (i = 0). Some of the correlations of planes of odd square order ($ t \neq 0 $ ) are also covered by the present analysis.When the companion automorphism is not trivial, the problem, naturally, becomes more involved, and a great deal begins to hinge upon the order of the plane being odd or even, and also a square or a nonsquare.The correlations of planes of order $ 2^{2^{i}(2n+1)}, n \neq 0 $, with companion automorphism $ 2^{2^{i}t}, t \neq 0 $, and especially those of planes of order $ p^{2^{i}(2n+1)}, i \neq 0 $, with companion automorphism $ p^{2^{j}(2r+1)}, j > i $ require a substantially different treatment, and will be the object of separate efforts.     

Convexity in finite metric spaces

A subsetS of a metric space (X,d) is calledd-convex if for any pair of pointsx,y S each pointz X withd(x,z) +d(z,y) =d(x,y) belongs toS. We give some results and open questions concerning isometric and convexity-preserving embeddings of finite metric spaces into standard spaces and the number ofd-convex sets of a finite metric space.     

On the generalized twisted field planes of characteristic 2

Let be a non-Desarguesian semifield plane of order 2 ⁿ 2⁶, and letG be the autotopism group relative to an autotopism triangle . We prove that ifG acts transitively on the non-vertex points on a side of , then is a generalized twisted field plane. A characterization of the generalized twisted field planes of characteristic 2 is also given.Research supported in part by NSF Grants RII-9014056, component IV of the EPSCoR of Puerto Rico grant and ARO grant for Cornell MSI.Research supported in part by NSF Grant No. DMS-9107372.     

The correlations of finite Desarguesian planes, Part III: The classification (II)

This paper continues the classification of the correlations of planes of odd nonsquare order. Part I (Generalities) included introductory definitions and results (Section 1), algebraic preliminaries (Section 2), as well as a discussion of equivalent correlations (Section 3) and of their general properties (Section 4). The classification proper revolves around a special polynomial which can have one, two, or q + 1 zeros, or no zeros at all, and each of these four possibilities leads to different families of correlations. Part II contained Section 5, devoted to the cases in which the correlation is defined by a diagonal matrix (Subsection 5.1) or the polynomial in the preceding paragraph possesses q + 1 zeros (Subsection 5.2), one zero (Subsection 5.3) and two zeros (Subsection 5.4). Subsection 5.5 presented certain results to be used in the subsequent sections. The present article contains Section 6, devoted to the case in which the above-mentioned polynomial has no zeros.     

The correlations of finite Desarguesian planes, Part II: The classification (I)

This paper continues the classification of the correlations of planes of odd nonsquare order. Part I (Generalities) – see reference [1]-included introductory definitions and results (Section 1), algebraic preliminaries (Section 2), as well as a discussion of equivalent correlations (Section 3) and of their general properties (Section 4). The classification proper revolves around a special polynomial which can have one, two, or q + 1 zeros, or no zeros at all, and each of these four possibilities leads to different families of correlations. The present article contains Section 5, devoted to the cases in which the correlation is defined by a diagonal matrix (Subsection 5.1) or the polynomial in the preceding paragraph possesses q + 1 zeros (Subsection 5.2), one zero (Subsection 5.3) and two zeros (Subsection 5.4). Subsection 5.5 presents certain results to be used in the subsequent sections.     

Covers of generalized quadrangles, 2. Kantor-Knuth covers and embedded ovoids

In this paper, which is a sequel to [12], we proceed with our study of covers and decomposition laws for geometries related to generalized quadrangles. In particular, we obtain a higher decomposition law for all Kantor-Knuth generalized quadrangles which generalizes one of the main results in [12]. In a second part of the paper, we study the set of all Kantor-Knuth ovoids (with given parameter) in a fixed finite parabolic quadrangle, and relate this set to embeddings of parabolic quadrangles into Kantor-Knuth quadrangles. This point of view gives rise to an answer of a question posed in [11].     

Laguerre planes of even order and translation ovals

Translation Laguerre planes of even order are represented in high dimensional projective space over GF(2) by a collection of subspaces that satisfies a very simple condition.This research was supported for the respective authors by a grant from the University of Canterbury and by a Feodor Lynen Fellowship.     

Subplanes of a tactical decomposition and singer groups of a projective plane

Sufficient and necessary conditions have been obtained for the following: (1) the substructure formed by a member of the partition of points and a member of the partition of lines to be a subplane; (2) the centralizer of a multiplier to be a Baer subplane. We establish the cyclicity of a Sylow 3-subgroup of the multiplier group of an abelian Singer group of square planar order. Sufficient conditions for the existence of a Type II divisor of a Singer group are given. For a Singer group of orderpq, p<q, we prove that if the order of the multiplier group is divisible byp, then the plane will admit a cyclic Singer group.Partially supported by a NSA grant