On Automorphisms of Flag Spaces

We show that the automorphisms of the flag space associated with a 3-dimensional projective space can be characterized as bijections preserving a certain binary relation on the set of flags in both directions. From this we derive that there are no other automorphisms of the flag space than those coming from collineations and dualities of the underlying projective space. Further, for a commutative ground field, we discuss the corresponding flag variety and characterize its group of automorphic collineations.     

On flag collineations of finite projective planes

s paper studies collineation groups of a finite projective plane containing flag collineations. Among other results, a characterization of a finite Desarguesian projective plane is given.Partially supported by grants from CNPq do Brasil and NSERC of Canada.     

Grassmann spaces of regular subspaces

The underlying metric affine geometry, or metric projective geometry, can be recovered from Grassmann spaces associated with the family of regular subspaces of respective space. In other words, automorphisms of such Grassmann spaces are collineations witch preserve orthogonality of the respective underlying space. This generalizes results of Pra?mowska et al. (Linear Algebra Appl 430:3066–3079, 2009) and Pra?mowska and ?ynel (Adv Geom, to appear).     

Canonical decomposition of projective and affine killing vectors on the tangent bundle

For an affine connection on the tangent bundle T(M) obtained by lifting an affine connection on M, the structure of vector fields on T(M) which generate local one-parameter groups of projective and affine collineations is described. On the T(M) of a complete irreducible Riemann manifold, every projective collineation is affine. On the T(M) of a projectively Euclidean space, every affine collineation preserves the fibration of T(M), and on the T(M) of a projectively non-Duclidean space which is maximally homogeneous (in the sense of affine collineations) there exist affine collineations permuting the fibers of T(M).Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 247–258, February, 1976.     

Transformations preserving adjacency and base subsets of spine spaces

Transformations of spine spaces which preserve base subsets preserve also adjacency. They either preserve the two sorts of projective adjacency or interchange them. Lines of a spine space can be defined in terms of adjacency, except one case where projective lines have no proper extensions to projective maximal strong subspaces, and thus adjacency preserving transformations are collineations.     

A family of flat Laguerre planes of Kleinewillingh?fer type IV.A

Just like Lenz–Barlotti classes reflect transitivity properties of certain groups of central collineations in projective planes, Kleinewillingh?fer types reflect transitivity properties of certain groups of central automorphisms in Laguerre planes. In the case of flat Laguerre planes, Polster and Steinke have shown that some of the conceivable types cannot exist, and they gave models for most of the other types. Only few types are still in doubt. Two of them are types IV.A.1 and IV.A.2, whose existence we prove here. In order to construct our models, we make systematic use of the restrictions imposed by the group generated by all central automorphisms guaranteed in type IV. With these models all simple Kleinewillingh?fer types with respect to Laguerre homologies and also with respect to Laguerre homotheties are now accounted for, and the number of open cases of Kleinewillingh?fer types (with respect to Laguerre homologies, Laguerre translations and Laguerre homotheties combined) is reduced to two.     

Collineations of polar spaces with restricted displacements

Let J be a set of types of subspaces of a polar space. A collineation (which is a type-preserving automorphism) of a polar space is called J-domestic if it maps no flag of type J to an opposite one. In this paper we investigate certain J-domestic collineations of polar spaces. We describe in detail the fixed point structures of collineations that are i-domestic and at the same time (i?+?1)-domestic, for all suitable types i. We also show that {point, line}-domestic collineations are either point-domestic or line-domestic, and then we nail down the structure of the fixed elements of point-domestic collineations and of line-domestic collineations. We also show that {i, i?+?1}-domestic collineations are either i-domestic or (i?+?1)-domestic (under the assumption that i?+?1 is not the type of the maximal subspaces if i is even). For polar spaces of rank 3, we obtain a full classification of all chamber-domestic collineations. All our results hold in the general case (finite or infinite) and generalize the full classification of all domestic collineations of polar spaces of rank 2 performed in Temmermans et?al. (to appear in Ann Comb).     

Partitions of Three-Dimensional Pappian Projective Spaces by Ovoidal Quadrics

In his article Partitioning Projective Geometries Into Caps, Ebert showed inter alia that every finite three-dimensional projective space can be partitioned by ovoidal quadrics. The aim of the present note is to generalize this result by identifying a more comprehensive class of three-dimensional projective spaces which admit partitions of this kind. In addition, the group of all collineations which preserve such a partition is determined.     

Morphisms of projective geometries and semilinear maps

Morphisms between projective geometries are introduced; they are partially defined maps satisfying natural geometric conditions. It is shown that in the arguesian case the morphisms are exactly those maps which in terms of homogeneous coordinates are described by semilinear maps. If one restricts the considerations to automorphisms (collineations) one recovers the so-called fundamental theorem of projective geometry, cf. Theorem 2.26 in [2].Supported by a grant from the Fonds National Suisse de la Recherche Scientifique.     

Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hähl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).     

On orthogonality-preserving plücker transformations of hyperbolic spaces

A complete overview of all orthogonality-preserving Plücker transformations in finite dimensional hyperbolic spaces with dimension other than three is given. In the Cayley-Klein model of such a hyperbolic space all Plücker transformations are induced by collineations of the ambient projective space.     

On secant dimensions and identifiability of flag varieties

We investigate the secant dimensions and the identifiability of flag varieties parametrizing flags of subspaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.     

Automorphisms of Fermat-like varieties

 We study the automorphisms of some nice hypersurfaces and complete intersections in projective space by reducing the problem to the determination of the linear automorphisms of the ambient space that leave the algebraic set invariant. Received: 18 December 2000 / Revised version: 22 October 2001     

Normisomorphismen und Normkurven endlichdimensionaler projektiver Desargues-Räume

In a finite dimensional desarguesian projective space the set of all points of intersection of homologous lines of two projective bundles of lines is called a non-degenerated (n. d.) normal curve, if the projective isomorphism is nondegenerated. Every frame determines a n. d. projective isomorphism of two bundles of lines called a normal isomorphism; every n. d. projective isomorphism of two bundles of lines is a normal isomorphism. A definition of osculating subspaces of a normal isomorphism is given and we show how the osculating subspaces can be constructed by using linear mappings. Simple examples show that there may be collineations fixing a n. d. normal curve but not fixing the osculating subspaces of the associated normal isomorphism. The set of osculating hyperplanes of a normal isomorphism is a n. d. normal curve in the dual space if and only if a certain number-theoretical condition holds.

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Herrn emer.O. Univ.-Prof. Dr. J. Krames zum 85. Geburtstag gewidmet     

Collineations of finite 2-affine planes

We investigate collineations of finite 2-affine planes and show that with the exception of some 2-affine planes of order at most 4 each collineation is induced by a collineation of its projective completion. We further deal in more detail with 2-affine planes of type II and their collineations that fix each long line.     

Conics in the hyperbolic plane intrinsic to the collineation group

In the manner of Steiner??s interpretation of conics in the projective plane we consider a conic in a planar incidence geometry to be a pair consisting of a point and a collineation that does not fix that point. We say these loci are intrinsic to the collineation group because their construction does not depend on an imbedding into a larger space. Using an inversive model we classify the intrinsic conics in the hyperbolic plane in terms of invariants of the collineations that afford them and provide metric characterizations for each congruence class. By contrast, classifications that catalogue all projective conics intersecting a specified hyperbolic domain necessarily include curves which cannot be afforded by a hyperbolic collineation in the above sense. The metric properties we derive will distinguish the intrinsic classes in relation to these larger projective categories. Our classification emphasizes a natural duality among congruence classes induced by an involution based on complementary angles of parallelism relative to the focal axis of each conic, which we refer to as split inversion (Definition 5.3).     

Geometry of free cyclic submodules over ternions

Given the algebra T of ternions (upper triangular 2×2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T ². This set of points can be represented as a set of planes in the projective space over F ⁶. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F-linear antiautomorphism, the plane model of our projective line does not admit any duality.     

Complex projective structures with maximal number of Möbius transformations

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Bely? curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms.     

Zentrale und axiale Kollineationen in projektiven Hjelmslev-Ebenen

Zusammenfassung Es gibt in projektiven Hjelmslev-Ebenen zentrale (axiale) Kollineationen, die keine Achse (Zentrum) haben. Das Produkt zweier zentraler Kollineationen s und t mit gemeinsamer Achse kann eine axiale Kollineation ohne Zentrum sein oder auch eine axiale Kollineation mit einem Zentrum, daß auf keiner Verbindungsgeraden der Zentren von s und t liegt.

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In ordinary projective planes every central collineation has an axis and every collineation with an axis is central. We prove in this paper, that this proposition doesn't hold in projective Hjelmslev-planes. We construct a projective Hjelmslev-plane and collineations with centers Pand Q on a common axis g such that the product of these collineations has no center but the axis g. In the dual plane we get a central collineation without an axis.

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Herrn R. Artzy zum siebzigsten Geburtstag gewidmet