Almost-Baer subplanes of free projective planes

We examine Baer subplanes, and subplanes which are almost-Baer subplanes, of any free plane—showing the subplanes are pairwise isomorphic—thus giving further examples of planes investigated by Barlotti.On leave from the University of Tasmania, Hobart, Australia.     

A Lie algebra attached to a projective variety

Abstract. Each choice of a K?hler class on a compact complex manifold defines an action of the Lie algebra sl(2) on its total complex cohomology. If a nonempty set of such K?hler classes is given, then we prove that the corresponding sl(2)-copies generate a semisimple Lie algebra. We investigate the formal properties of the resulting representation and we work things out explicitly in the case of complex tori, hyperk?hler manifolds and flag varieties. We pay special attention to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra. Oblatum 21-V-1996 & 15-X-1996     

A hypergroupal characterization of projective planes

We give two sufficient conditions for a hypergroupoid to be a feeble semi-hypergroupoid and a sufficient condition to be a feeble hypergroup. We show that every hypergroup in the class of the k-quasi-Steiner hypergroupoids is feebly associative. Finally, we apply these algebraic results to study Steiner systems; in particular, we give a necessary and sufficient condition for a Steiner system to be a projective plane.     

Pathological projective planes: Associate affine planes

In [7] the author showed the existence of projective plane pathological with respect to the collineation groups of its sub and quotient planes. Similar pathologies are obtainable with respect to collineation groups of associated affine planes. (i.e. the affine planes obtained by distinguishing a line as the line at infinity) as expressable in the following theorem.     

Stable periodic projective planes

We find all stable projective planes with finite topology which are properly embedded in , where is a discrete subgroup of translations in . Here stable means second order minimum of the area. The surfaces we obtain are a quotient of the helicoid and quotients of the doubly periodic Scherk surfaces.

     

Constructively presented projective planes

An attempt to define a finite presentation for projective planes by analogy to finitely presented groups has been made in [9]. In this article we suggest another definition for a presentation of projective planes: constructively presented planes. A few simple results about such planes and an example of constructively presented planes with nonsolvable word problem are given.This paper was written while the author was visiting the University of Bergen, Norway.     

Minimal topological projective planes

The atoms of the lattice of compatible topologies on a given projective plane are examined. The notion of a field of type V is generalized to ternary fields of type V. These are always minimal, and they arise for example as coordinate structure of strictly uniformizable or of orderable topological planes. Hence, such planes are minimal.Dedicated to Professor Helmut Karzel on his 60^th birthday     

Gorenstein Q-homology projective planes

We present the complete list of all singularity types on Gorenstein Q-homology projective planes,i.e.,normal projective surfaces of second Betti number one with at worst rational double points.The list consists of 58 possible singularity types,each except two types supported by an example.     

Smooth projective translation planes

A projective plane is called smooth if both the point space and the line space are smooth manifolds such that the geometric operations are smooth. We prove that every smooth projective translation plane is isomorphic to one of the classical planes over , , or .Dedicated to Professor Dr. H. Salzmann on the occasion of his 65th birthday     

Cartesian groups not belonging to topological projective planes

A new class of topological Cartesian groups is introduced by bending the lines of the affine coordinate plane over an ordered skewfield countably infinitely often. These Cartesian groups belong to topological affine planes with continuous parallelism, but they need not yield a topological projective plane if the topology is not equal to the order topology of the skewfield, in contrast to the results of HARTMANN [6] for generalized Moulton planes. Dedicated to Professor Dr. Gerhard Grimeisen on his 70 ^th birthday