Małgorzata Prażmowska

Multiple perspectives and generalizations of the Desargues configuration

We introduce a class of finite configurations, which we call combinatorial Grassmannians, and which generalize the Desargues configuration. Fundamental geometric properties of them are established, in particular we determine their automorphisms, correlations, mutual embedability, and prove that no one of them contains a Pascal or Pappus figure. Introduction The classical Desargues configuration, nearly for ages, was a source of mathematical investigations. Its role in foundations of projective geometry is well known (see [9] or, more recent, [6]). Some of its special forms were studied in finite geometries (see e.g. [3]). Interesting questions concerning various realizations of this configuration in a projective plane and the groups of projective collineations of these realizations were discussed in the beautiful paper [2]. In this paper we discuss some other generalization of the Desargues configuration which originates in investigations concerning combinatorial structures. Our incidence structures are partial linear spaces and will be called combinatorial Grassmannians; one can say that they generalize the Desargues configuration to higher dimensions (cf. [8] quoted in 1.2). The definition of a combinatorial Grassmannian & can easily be expressed in terms of elementary Combinatorics of finite sets: the points of <5 are the A;-element subsets of a fixed set X, for some fixed integer k, the blocks are the (k + l)-subsets of X, and the incidence of (25 is the inclusion. It is interesting to note that, as the Desargues configuration represents the perspective of two triangles, a combinatorial Grassmannian is an abstract schema of multiple perspective of a finite number of simplices (cf. 1.7); this fact indicates that our structures really may have something in common with geometry. In the paper, however, we do not pay too much attention to (projective) representation of combinatorial Grassmannians. Instead, we are concentra-

Twisted Fano spaces and their classification, linear completions of systems of triangle perspectives

Twisted Fano spaces i.e. linear spaces with the parameters of PG(3, 2) which contain a pencil of Fano subplanes are completely classified and characterized. In particular, it is proved that twisted Fano spaces are exactly all the linear completions of systems of triangle perspectives with point degree 4.

103-configurations and projective realizability of multiplied configurations

Some remarks on 103-configurations which contain the complete graph K4 are given, on their representations, and on projective realizability. Results are applied to show a class of configurations that cannot be realized in any Pappian projective space.

A complete classification of the ( 15 4 20 3 ) -configurations with at least three K 5 -graphs

The class of ( ( n + 1 2 ) n - 1 ( n + 1 3 ) 3 ) -configurations which contain at least n - 2 K n -graphs coincides with the class of so called systems of triangle perspectives i.e.?of configurations which contain a bundle of n - 2 Pasch configurations with a common line. For n = 5 the class consists of all binomial partial Steiner triple systems on 15 points, that contain at least three K 5 -graphs. In this case a complete classification of respective configurations is given and their automorphisms are determined.

Combinatorial generalizations of generalized quadrangles of order (2, 2)

We study substructures of a projective space PG(n, 2) represented in terms of elementary combinatorics of finite sets, which generalize the Sylvester’s representation of the generalized quadrangle of order (2, 2). Their synthetic properties are established and automorphisms are characterized.

Desarguesian closure of binomial graphs

In the paper we study configurations which are obtained as Desarguesian closure of binomial graphs. Their parameters are calculated, and their automorphisms are determined.

On some regular multi-veblen configurations, the geometry of combinatorial quasi Grassmannians

Multi-Veblen configurations which can be embedded into Desarguesian projective spaces were characterized in [8]: besides the class of combinatorial Grassmannians, only one further class of multi-Veblen configurations shares this property, namely the class of combinatorial quasi Grassmannians introduced in [8]. In this note we discuss relationships between combinatorial Grassmannians and combinatorial quasi Grassmannians, characterize automorphisms of combinatorial quasi Grassmannians, and present some visualizations of them. Introduction There is a great deal of a literature on Steiner triple systems, and the literature on partial (called also incomplete) Steiner triple systems is also growing rapidly. The subject of this note is the geometry of some particular partial Steiner triple systems; why does this particular class deserve our interest? In our opinion, we can quote two reasons. First, the structures considered in the paper are multi-Veblen configurations. The class of multi-Veblen configurations was introduced in [10]. These configurations arise naturally as combinatorial schemes generalizing two classical configurations considered in geometry, namely Kantors IO3G configuration, and the Desargues configuration, presented as some systems of Veblen configurations. Secondly, the structures considered in the paper are projectively embeddable i.e. they can be embedded into Desarguesian projective spaces. It turns out (cf. [8] for details) that there are exactly two types of multi-Veblen configurations that can be projectively embedded. Structures of the first type are so called

Semi-Pappus configurations; combinatorial generalizations of the Pappus configuration

A class of partial Steiner triple systems generalizing a reduct of the classical Pappus configuration and thus called semi-Pappus configurations is defined. Fundamental geometric properties, in particular, representations and the automorphisms of semi-Pappus configurations and of the convolutions of semi-Pappus configurations and the group C2 are established.

A proof of the projective Desargues axiom in the Desarguesian affine plane

We give a short proof that the projective Desargues axiom is valid in the Desarguesian affine planes.

An axiom system describing degenerate hyperbolic planes in terms of directed parallelity

Dedicated to the memory of Professor Edward Otto Introduction In this papei we give an axiom system foi the class oi degeneiate hyperbolic planes and discuss some features of these structures. Usual, clas.sical hyperbolic plane is a structure whose universe consists of points inside some fixed conic in a projective plane (c.f. (11). In analogy we call a degenerate hyperbolic plane a structure in which point-universe is a interior of degenerate conic and lines are appropriate secants. Such a structure can be visualized as an open affine half-plane or (which is essentially the same) as an affine open stripe with parallel boundary lines. The structure of collineation group of degenerate hyperbolic planes was investigated in 16], 17] and 15); several sufficient systems of primitive notions were shown in 15). The language we choose contains as a primitive notion the relation of directed parallelity; in the affine model such a relation means that appropriate half lines meet at the horizon,