Dual Affine Plane Research Articles

Characterizations of Translation GeneralizedQuadrangles

If x is a regular point of the generalized quadrangle \mathcal{S} of order (s,t), s\ne 1 \ne t, then x defines a dual net \mathcal{N}^\ast_x. If \mathcal{S} contains a line L of regular points and if for at least one point x on L the automorphism group of the dual net \mathcal{N}_x^\ast satisfies certain transitivity properties, then \mathcal{S} is a translation generalized quadrangle. This result has many applications. We give one example. If s=t\ne 1, then \mathcal{N}^\ast_x is a dual affine plane. Let \mathcal{S} be a generalized quadrangle of order s,s odd and s\ne 1, which contains a line L of regular points. If for at least one point x on L the plane \mathcal{N}^\ast_x is Desarguesian, then \mathcal{S} is isomorphic to the classical generalized quadrangle W(s).

Giuseppe Veronese and Ernst Witt—neighbours in PG(5,3)

Let $P$ be a point of the Veronese surface $\Vcal$ in \PG53. Then thereare four conics of $\Vcal$ through $P$. We show that the internal points of those conics form a 12-cap which is a point model for Witt's 5-$(12,6,1)$ design. In fact, this construction is to a similar construction that has been established by the author. We give an explicit parametrization of the cap $\Kcal$; the domain is a dual affine plane which arises from \PG23 by removing one point. Thus, as a by--product, we obtain an easy approach to the extended ternary Golay code $G_{12}$. Finally, we discuss some other procedures that yield 12-sets of points from the Veronese surface.

Giuseppe Veronese and Ernst Witt—neighbours in PG(5,3)

Let P be a point of the Veronese surface $ \nu $ in PG(5,3). Then there are four conics of $ \nu $ through P. We show that the internal points of those conics form a 12-cap which is a point model for Witt's 5-(12,6,1) design. In fact, this construction is “dual” to a similar construction that has been established in [6] recently. We give an explicit parametrization of the cap $ {\cal K} $ the domain is a dual affine plane which arises from PG(2,3) by removing one point. Thus, as a by-product, we obtain an easy approach to the extended ternary Golay code $ G_{12} $ . Finally, we discuss some other procedures which yield 12-sets of points from the Veronese surface.

Geometries of type [L.Af*

Let Γ be a [L.Af*]-geometry, that is a rank 3 geometry with linear spaces as plane residues, with dual affine planes as point residues and with generalized digons as line residues. Assume that (LL) holds in Γ. In the particular case where the plane residues are finite circles, the structure of such geometries has been strongly restricted by A. P. Sprague. Moreover, C. Lefevre and L. Van Nypelseer have given a complete classification of such geometries under the assumption that the plane residues are affine planes. We generalize these two results for [L.Af*]-geometries.

On delta spaces satisfying Pasch's axiom

We consider partial linear spaces all of whose lines contain at least three points and in which every pair of intersecting lines generates a subspace isomorphic to a projective or dual affine plane. In particular, we classify in this paper those spaces that contain a projective plane.

Af∗ . Af geometries, the Klein quadric and [formula omitted]nq

An Af ∗. Af geometry of order q is a residually connected rank three geometry where planes are dual affine planes and stars of points are affine planes of order q. We prove that such a geometry is necessarily obtained from the Klein quadric Q + 5( q) of PG(5, q) deleting the points of a hyperplane and considering as points the elements of one of the two systems of maximal subspaces of Q= Q + 5( q), as lines the points of Q, and as planes the elements of the other system. The deleted hyperplane is tangent to Q if and only if the Af ∗. Af geometry obtained satisfies property (PL 1) (i.e. there is a unique plane on every point-line antiflag). When (PL 1) is satisfied, some generalizations are obtained for L ∗ . L (resp. N ∗. L) geometries (i.e. residually connected rank three geometries where planes are dual linear spaces (resp. dual nets) and stars of points are linear spaces). In particular, this yields a characterization of H n q and T ∗ 2( K), where K = AG(2, q), in the context of rank 3 partial geometries. Furthermore, it leads to some classification results for other rank n⩾3 diagrams related to what we call the rank n H n q ( n⩾3, see Examples 5.1 and 5.4 for the definition).

The Steiner system S(5,8, 24) constructed from dual affine planes

SynopsisWe construct firstly a single tactical configuration which has the structure of the dual of the affine plane of order 4, and show how to obtain a further set of 3 such dual planes which, together with , satisfy a certain set of intersection properties. This set of 4 dual planes is used to extend the 20 points of to the Steiner system = S(5, 8, 24). The construction leads to the production of involutions of the type which fix the points of an octad. It is shown that 3 involutions each of this type suffice to generate M24, each of the simple Mathieu groups inside M24, the Todd group, and all the intransitive maximal subgroups of M24.

A characterization of special laguerre planes and extended dual affine planes

A special Laguerre plane is a nondegenerate transversal 3-design such that the residue of each point is a dual affine plane. A special Laguerre plane is equivalent to an optimal code with three information digits and maximal length. An extended dual affine plane is an incidence structure (whose objects will be called points and blocks) such that the residue of each point is a dual affine plane, and each pair of points is in at least one block. Finite extended dual affine planes exist only of order 2, 4, and (dubiously) 10. We show that any finite incidence structure having the residue of each point a dual affine plane either is a transversal 3-design or has a block through each pair of points. Hence theorem: If a finite nondegenerate connected incidence structure θ has the residue of each point a dual affine plane, then θ is either an extended dual affine plane or a special Laguerre plane.

Extended dual affine planes

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.

Geometries with dual affine planes and symplectic quadrics

Farmer and Hale [3] prove that every copolar space fully embedded in a finite projective space PG( n, q), with q>, is the copolar space arising from a symplectic polarity. We show that this result is still valid in arbitrary projective spaces; this provides a different and shorter proof of [3] in the finite case.

Dual affine geometries and alternative bilinear forms

The classical projective and affine geometries can be described as geometries in which each plane is a Desarguesian projective or affine plane. The principal theorem provides an analogous description for geometries induced on projective space by an alternative bilinear form, which are called symplectic geometries: Let G be a geometry in which each plane is a dual affine plane over a field of q elements, q >2, and suppose that the points and lines of G are points and lines generating some projective space P . Then the lines of G are images of nonisotropic 2-spaces with respect to an alternative bilinear form on the vector space from which P was derived.

Finite geometries which contain dual affine planes

Finite geometries in which each plane is projective or dual affine over the field of two elements, or affine over the field of three elements, are studied. It is shown that no connected geometry can mix all three species of planes, and the geometries in which projective and dual affine planes occur are classified.