4-Dimensionale projektive Ebenen mit grosser abelscher Kollineationsgruppe

We show that a 4-dimensional connected abelian group can act in exactly five different ways as a collineation group of a compact 4-dimensional projective plane. Furthermore the complex projective plane is characterized as the only compact 4-dimensional projective plane which admits two different 4-dimensional abelian collineation groups.

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Herrn Professor Dr. Eelmut Karzel zum 60. Geburtstag     

Projektive Abschliessbarkeit von Kinematischen Räumen

Kinematic spaces of an order greater than 3 with only affine or projective lines are slit spaces i.e. trace spaces of projective spaces where a subspace is omitted.

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Herrn Professor Dr. Helmut Karzel zum 50. Geburtstag     

Ein “affiner” Beweis des Satzes von v. Staudt—Schleiermacher

The Theorem ofv. Staudt-Schleiermacher asserts that a projective incidence plane is pappian if every projectivity with 5 fixed points is the identity.Schleiermacher's original proof needs different treatments for a lot of cases. We give a simpler proof based on the fact that Desargues' and Pappos' theorems hold in a projective plane if this is true for some affine specializations.

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Mit 8 Abbildungen     

Zur algebraischen Kennzeichnung mehrwertiger Halbordnungen

The radical R(T) of a planar ternary ring (PTR) T, which plays an important role in the theory of halforderings and orderings of PTRs, allows also to give an algebraic characterization ofJunkers' multiple-valued halforderings of projective planes. As an application we obtain some information on the group of projectivities in projective planes.

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Herrn Professor G. Tallini zum 60. Geburtstag gewidmet     

über Kurven vom TYP (m;n) und Beispiele total m-regulärer (k,n)-Kurven

In this paper we are concerned with a special kind of subsets of finite projective planes and give some new examples of totally m-regular (k,n)-arcs.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

Eine Bemerkung über den speziellen Bezoutschen Satz

Some synthetic methods in a projective space are devellopped and using them a theorem connected with the theorem of Bézout is proved without using any postulation about algebraic closure.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

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     

Una proprietà gruppale delle involuzioni planari che mutano in sé un'ovale di un piano proiettivo finito

Summary The purpose of the present paper is to prove the following theorem: Let Ω be an oval in the projective plane P of odd order n. If P admits a collineation group G wich maps Ω onto itself and is doubly transitive on Ω, then P is desarguesian, Ω is a conic and G contains all collineations in the little projective group PSL(2, n) of P wich leaves Ω invariant.

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Entrata in Redazione il 5 april 1977.     

Die windschiefen Flächen mit einer stetigen Schar ebener Schattengrenzen I

Already in 1933J. Blank found all ruled surfaces with two conjugated families of plane shade lines. In the late sixtiesH. Brauner solved a similar problem by determining all algebraic surfaces with only one family of plane shade lines. — We consider here only differentiable ruled surfaces withBrauner's condition, that carry at least one continuous family of plane shade lines. By using methods of projective differential geometry, algebra and synthetic projective geometry, it is possible to find all projective representatives of those surfaces. Although the assumption ofBrauner is much weaker thanBlank's, each of these surfaces is an analytic, ruled surface ofBlank.

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Mit 2 Abbildungen     

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     

Projektive Inzidenzgruppoide und Fastalgebren

In this paper we extend the notion of an incidence group to that of a groupoid. The concept of near-algebras, introduced below, enables us to describe Desarguesian incidence groupoids algebraically and to generalize most of the essential results about projective incidence groups with regard to groupoids.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

Halbordnungstopologien in Projektiven Ebenen

The order topology of an ordered projective plane leads to a topological projective plane in which intersection and joining are continuous operations (c.f. O.WYLER [5]).Here continuity properties in projective planes with half order topology derived from a SPERNER'sche Ordnungsfunktion are analysed. Central projections are continuous and planes which are connected with respect to the half order topology are (fully) ordered and therefore topological planes.

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Herrn Prof. Dr. WERNER BURAU zum 70. Geburtstag     

Vers un algorithme pour la réduction stable des revêtements <Emphasis Type="Italic">p</Emphasis>-cycliques de la droite projective sur un corps <Emphasis Type="Italic">p</Emphasis>-adique

 In his Ph. D. thesis, C. Lehr offers an algorithm which gives the stable model for p-cyclic covers of the projective line over a p-adic field under the conditions that the branch locus whose cardinal is m+1 has the so called equidistant geometry and m<p. In this note we give an algorithm also in the equidistant geometry case but without condition on m. In particular we are able to study the reduction at 2 of hyperelliptic curves with equidistant branch locus.

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Vers un algorithme pour la réduction stable des revêtements p-cycliques de la droite projective sur un corps p-adique

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Received: 11 February 2002 / Revised version: 8 May 2002 / Published online: 2 December 2002

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Mathematics Subject Classification (2000): 11G20, 14H30, 14Q05     

Zur Fortsetzbarkeit von Ordnungsfunktionen einer affinen Ebene

We show that every order function of a translation plane with kernel GF(2), GF(3) can he extended to an order function of the projective closure.

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Herrn Professor Dr. Hanfried Lenz zum 70. Geburtstag     

Über fastgeordnete affine Ebenen

An example of an affine plane is constructed, which cannot be ordered, but can be embedded in an ordered projective plane. There is no such an example, if the affine plane is a translation plane with kernel GF(2).

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Herrn Rafael Artzy zum 75. Geburtstag gewidmet     

Doppelebenen und Loops

Generalizing the concept of difference sets in groups conditions are given for a loop (P, +) (with equality of left and right inverses) and a subset D of P such that (i) the left translates of D and the right translates of -D (the set of inverses for elements of D) resp. are the lines of two projective planes with point set P, forming a double plane (i.e. the lines of one plane are ovals of the other), (ii) the loop operation has a certain geometric interpretation in the double plane.

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Walter Benz zum 60. Geburtstag     

Jordan-Basen projektiver Räume bezüglich linearer Selbstabbildungen

In vector spaces the notion of Jordan base is useful in considerations of the Jordan normal form of matrices. The notion of Jordan base with respect to a given linear self-mapping of a projective space is defined, and the existence of such a Jordan base is demonstrated under weak assumptions about .

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Herrn Prof. Dr. Walter Wunderlich zum 80. Geburtstag gewidmet     

Groupe de Chow de codimension deux des variétés définies sur un corps de nombres: un théorème de finitude pour la torsion

Summary LetX be a smooth, projective variety defined over a number field and let CH² (X) denote the Chow group of codimension two cycles modulo rational equivalence. We show that if the cohomology groupH ²(X,O_x) vanishes then the torsion subgroup of CH² (X) is a finite group. This result covers all previous results in this direction. The hypothesisH ²(X,O_x)=0 is used to lift line bundles.

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Oblatum 17-IX-1990     

Un phénomène de concentration de genre

We show that the main part of the genus of the preimages of a projective line by a generic holomorphic endomorphism of goes to the support of the Green measure.

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Résumé Nous montrons que le genre des préimages dune droite projective par un endomorphisme holomorphe générique de se concentre sur le support de la mesure de Green.A phenomenon of concentration of genus

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Mathematics Subject Classification (2000): 32H50, 32U40, 32Q45     

Über die Projektivitätengruppen von Semikörperebenen gerader Ordnung

Let S be a finite proper semifield of even order and II the group of projectivities of the projective plane over S.We prove that if S is commutative or of Knuth-type II, III or IV and ¦S¦>16 then consists of even permutations only. On the other hand we give examples of semifields of order 16, 32 and 64 whose corresponding groups contain odd permutations.

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Dieser Artikel entstand als Kurzfassung einer von Hr. Prof. Dr. J. Joussen vergebenen Diplomarbeit.

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Nicht nur für die Betreuung während der Diplomarbeit, sondern auch für die Hilfe bei der Erstellung dieser Kurzfassung, bin ich Hr. Prof. Dr. J. Joussen sehr dankbar.