arXiv subjects
Publications and source records attributed to K. Prażmowski.
We study parallelisms on Veronese spaces associated with affine spaces, determine hyperplanes in Veronese spaces associated with projective spaces, and analyse the geometry of parallelisms determined by these hyperplanes.
We propose a new approach to studies on partial Steiner triple systems consisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product we introduce the notion of a binomial PSTS as a configuration with parameters of a minimal PSTS with a complete subgraph. A representation of binomial PSTS with at least a given number of its maximal complete subgraphs is given in terms of systems of perspectives. Finally, we prove that for each admissible integer there is a binomial PSTS with this number of maximal complete subgraphs.
We propose a generalization of the classical point-line Cremona-Richmond configuration to a configuration of points and more dimensional subspaces of a projective space, and present them as geometric realizations of some interesting abstract incidence structures.
The class of $\left(\binom{n+1}{2}_{n-1} \binom{n+1}{3}_3\right)$-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.
The realizability of countable Cox configurations on Miquelian planes is proved. A simple way to determine the automorphisms of the Cox configurations is presented.
We prove that an affine polar space in the meaning of Cohen and Shult can be recovered from one of the three adjacency relations on a Grassmann structure over it. The result directly generalizes the results of our previous work where we use an affine space over a vector space equipped with a nondegenerate reflexive form as a starting point to the Cohen-Shult affine polar spaces.
We construct Grassmann spaces associated with the incidence geometry of regular and tangential subspaces of a symplectic copolar space, show that the underlying metric projective space can be recovered in terms of the corresponding adjacencies on so distinguished family of k-subspaces (geometrical dimension of the space being not 2k+1), and thus we prove that bijections which preserve the adjacency are determined by automorphisms of the underlying space.
The regular point-line geometry with respect to a pseudo-polarity is introduced. It is weaker than the underlying metric-projective geometry. The automorphism group of this geometry is determined. This geometry can be also expressed as the geometry of regular lines and planes.
Deleting a hyperplane from a polar space associated with a symplectic polarity we get a specific, symplectic, affine polar space. Similar geometry, called an \afsempol\ arises as a result of generalization of the notion of an alternating form to a semiform. Some properties of these two geometries are given and their automorphism groups are characterized.