Armin Herzer

DIE SCHMIEGHYPEREBENEN AN DIE VERONESE-MANNIGFALTIGKEIT BEI BELIEBIGER CHARAKTERISTIK

By means of linear algebra a base-free definition of a Veronese variety V(n,r) is given and also an illuminating description of its osculating primes from which can be deduced in a general form and without difficulty the phenomena of degeneracy in case of small characteristics. (Instance best known: For characteristic 2 all tangents of a conic are confluent.)The last section investigates special problems for the V(1,r) in characteristic p: So the osculating primes of a V(1,p) intersect its node in a V(1,p-2). Furthermore it becomes clearer why for 2<r<¦K¦−1 no elation can fix a V(1,r) (in case of a perfect field).

On sets of subspaces closed under reguli

Using a representation of chain geometries where points are certain subspaces of a projective space and chains are reguli, we give an algebraic description of the weak subspaces of the chain geometry (i.e. the subsets of the pointset which are closed with respect to reguli).

Characterization of chain geometries of finite dimension by their automorphism group

A large class of chain geometries of finite dimension is characterized as strong chain spaces possessing a distinguished group of automorphisms fixing two distant points.

Charakterisierung von Kettengeometrien über semilokalen Algebren

Chain geometries over semilocal rings are characterized as special touching structures possessing a distinguished group of automorphisms fixing two distant points.

Der Satz von Tits für PGL2(R), R ein kommutativer Ring vom stabilen Rang 2

Certain permutation groups on sets with distance relation are characterized as groups of projectivities PGL2(R) on the projective line over a commutative ring R of stable rank 2, thus generalizing a classical result of Tits where R is a field.

Characterization of strong chain geometries by their automorphism group

A wide class of chain geometries is characterized by their automorphism group using properties of a distinguished involution.

Kennzeichnung von Berührstrukturen, die Kettengeometrien sind

Chain geometries already were characterized by means of chain spaces possessing a distinguished automorphism group. Here instead of this the weaker concept of contact spaces is used.

Affine Kettengeometrien über Jordanalgebren

It is shown that an affine chain geometry over a Jordan algebra can be constructed in a nearly classical manner. Conversely, such chain geometries are characterized as systems of rational normal curves having a group of automorphisms with certain properties.

On Finite Translation Structures with Proper Dilatations

Recently, Biliotti and the author obtained a certain number of results on translation structures with proper dilatations including structure-and characterisation-theorems, which here will be reformulated in a different manner, throwing a new light on some of the regarded questions.

Ein Satz von l. Schur über vertauschbare matrizen

Abstract We give a new proof of a theorem of I. Schur, describing all commutative subalgebras of maximal dimension of the matrix ring n .