Flat projective planes with 2-dimensional collineation group fixing at least two lines and more than two points

All flat projective planes whose collineation group contains a 2-dimensional subgroup fixing at least two lines and more than two points are classified. Furthermore, all isomorphism types of such planes are determined. This completes the classification of all flat projective planes admitting a 2-dimensional collineation group.     

4-Dimensional compact projective planes with a 5-dimensional nilradical

The article is a contribution to the classification of all 4-dimensional flexible compact projective planes. We assume that the collineation group is a 6-dimensional solvable Lie group which fixes some flag. If, moreover, the nilradical of the collineation group is 5-dimensional, then we get 4 families of new planes which are neither translation planes nor shift planes.Meinem Lehrer H. Salzmann zum 65. Geburtstag am 3.11.1995 in Dankbarkeit gewidmet     

Four-Dimensional Compact Projective Planes of Orbit Type (1,1)

We consider 4-dimensional flexible projective planes with the following properties: The collineation group is a 6-dimensional solvable Lie group which fixes some flag ∞ ∈ W. Furthermore, the collineation group has a 1-dimensional orbit both on W and on the pencil of lines through {∞}. We show that there are three different families of planes with these properties.     

Die Oktavenebene als Translationsebene mit großer Kollineationsgruppe

It is shown that the affine plane over the Cayley numbers is the only 16-dimensional locally compact topological translation plane having a collineation group of dimension at least 41. This (hitherto unpublished) result is one of the ingredients of H. Salzmann's characterizations of the Cayley plane among general compact projective planes by the size of its collineation group.The proof involves various case studies of the possibilities for the structure and size of collineation groups of 16-dimensional locally compact translation planes. At the same time, these case studies are important steps for a classification program aiming at the explicit determination of all such translation planes having a collineation group of dimension at least 38.     

Polaritäten Ebener projektiver Ebenen

A projective plane is called flat if the spaces of points and lines are locally compact and 2-dimensional and the joining of points and the intersecting of lines are continuous. H. Salzmann studied planes of this type in [11]–[21]. Here polarities of such planes are considered. In II general properties of polarities of flat planes are discussed. For example, a polarity with absolute points has always an oval of absolute points. A flat projective plane with a cartesian ternary field K admits a polarity iff multiplication in K is commutative. In III the polarities of flat projective planes with a 3-dimensional collineation group are determined.     

16-Dimensional Smooth Projective Planes with Large Collineation Groups

Smooth projective planes are projective planes defined on smooth manifolds (i.e. the set of points and the set of lines are smooth manifolds) such that the geometric operations of join and intersection are smooth. A systematic study of such planes and of their collineation groups can be found in previous works of the author. We prove in this paper that a 16-dimensional smooth projective plane which admits a collineation group of dimension d 39 is isomorphic to the octonion projective plane P₂ O. For topological compact projective planes this is true if d 41. Note that there are nonclassical topological planes with a collineation group of dimension 40.     

4-dimensional compact projective planes with a 7-dimensional collineation group

The classification of 4-dimensional compact projective planes having a 7-dimensional collineation group is completed. Besides one single shift plane all such planes are either translation planes or dual translation planes.Dedicated to H. R. Salzmann on his 60th birthday     

Projektive Darstellung der Moulton-Ebenen

The Moulton planes can be characterized as 2-dimensional topological projective planes having a 4-dimensional collineation group, which fixes exactly one nonincident point-line-pair aw. We give a representation of these geometries on the real protective plane such that a and W coincide with the origin and the line of infinity. This representation shows that the collineation groups of nonisomorphic Moulton planes act differently, although they are isomorphic as topological groups.     

Elementary constructions of locally compact affine planes

In this paper we describe several elementary constructions of 4-, 8- and 16-dimensional locally compact affine planes. The new planes share many properties with the classical ones and are very easy to handle. Among the new planes we find translation planes, planes that are constructed by gluing together two halves of different translation planes, 4-dimensional shift planes, etc. We discuss various applications of our constructions, e.g. the construction of 8- and 16-dimensional affine planes with a point-transitive collineation group which are neither translation planes nor dual translation planes, the proof that a 2-dimensional affine plane that can be coordinatized by a linear ternary field with continuous ternary operation can be embedded in 4-, 8- and 16-dimensional planes, the construction of 4-dimensional non-classical planes that admit at the same time orthogonal and non-orthogonal polarities. We also consider which of our planes have tangent translation planes in all their points. In a final section we generalize the Knarr-Weigand criterion for topological ternary fields.This research was supported by a Feodor Lynen fellowship.     

Multiplicative Loops of 2-Dimensional Topological Quasifields

We determine the algebraic structure of the multiplicative loops for locally compact 2-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a 1-dimensional compact subgroup. In the last section, we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension 4 admitting an at least 7-dimensional Lie group as collineation group.     

Connected 4-dimensional stable planes with many central collineations

This paper gives a complete classification of all connected 4-dimensional stable planes with the property that every point is the centre of a non-trivial central collineation. It is shown that under these assumptions the automorphism group has an open orbit on the point space. This implies the existence of an open subplane that carries the additional structure of a generalized symmetric space in the sense of differential geometry. Now the classification of all 4-dimensional generalized symmetric planes yields the desired classification.Dedicated to Professor H. Salzmann on his 60th birthday     

Geometry of curves with exceptional secant planes: linear series along the general curve

We study linear series on a general curve of genus g, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill?CNoether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of d-secant (d ? 2)-planes to (2d ? 1)-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in d.     

Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout-Metz unitals

It is shown that for every semifield spread in PG(3,q) and for every parabolic Buekenhout-Metz unital, there is a collineation group of the associated translation plane that acts transitively and regularly on the affine points of the parabolic unital. Conversely, any spread admitting such a group is shown to be a semifield spread. For hyperbolic Buekenhout unitals, various collineation groups of translation planes admitting such unitals and the associated planes are determined.     

On 4-dimensional Minkowski planes with 7-dimensional automorphism groups

This paper concerns 4-dimensional (topological locally compact connected) Minkowski planes that admit a 7-dimensional automorphism group . It is shown that such a plane is either classical or has a distinguished point that is fixed by the connected component of and that the derived affine plane at this point is a 4-dimensional translation plane with a 7-dimensional collineation group.This research was supported by a Feodor Lynen Fellowship.     

Construction of two-dimensional flag-transitive planes

An affine plane is called flag-transitive if it admits a collineation group which acts transitively on the incident point-line pairs. It has been shown that finite flag-transitive planes are necessarily translation planes, and much work has been devoted to this class of translation planes in recent years. All flag-transitive groups of finite affine planes have been determined, and an infinite family of non-Desarguesian flag-transitive planes has been found. In this paper a method is given for constructing all two-dimensional flag-transitive planes of odd order, subsuming the infinite family mentioned above.     

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     

Eine Klasse von achtdimensionalen lokalkompakten Translationsebenen mit großen Scherungsgruppen

This paper is one of the final steps in a classification program to determine all eight-dimensional, locally compact translation planes having large collineation groups. Here, we describe all such planes whose collineation group contains a semidirect product ·N, whereN is an at least 3-dimensional normal subgroup consisting of shears with fixed axis, and is isomorphic to SO₃ ().     

Projective planes of order 12 do not have a collineation group of order 4

In this paper, we prove that there are no projective planes of order 12 admitting a collineation group of order 4. This yields that the order of any collineation group of a projective plane of order 12 is 1, 2, or 3.