Restricted homogeneous coordinates for the Cayley projective plane

I. Porteous has shown that the Cayley projective plane can be coordinatized in a way resembling homogeneous coordinates. We will show how to construct line coordinates in a similar way. As an illustration, we give an explicit example to show that the Cayley projective plane is not Desarguean.     

On delta spaces satisfying Pasch's axiom

We consider partial linear spaces all of whose lines contain at least three points and in which every pair of intersecting lines generates a subspace isomorphic to a projective or dual affine plane. In particular, we classify in this paper those spaces that contain a projective plane.     

Stable projective planes with Riemannian metrics

We consider the question whether the system of lines of a two-dimensional stable plane can be described as the system of geodesics of a Riemannian metric and vice versa; we present two results: A complete two-dimensional Riemannian manifold with the property that every two points are joined by a unique geodesic and its family of geodesics form a stable plane. On the other hand every stable projective plane whose lines are geodesics of a Riemannian metric is isometric to the real projective plane. Combining both results it follows that it is impossible to realize the lines of a non-desarguesian projective plane using the geodesics of a complete Riemannian manifold.     

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     

Galois lines for the Giulietti–Korchmáros curve

We describe the arrangement of all Galois lines for the Giulietti–Korchmáros curve in the projective 3-space. As an application, we determine the set of all Galois points for a plane model of the GK curve. This curve possesses many Galois points.     

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.     

Projective planar spaces

A local condition on a planar space is given which is sufficient for its points, lines and planes to be the points, the lines and some subspaces of a projective space.     

Algebraic properties of grids of projective lines

In this paper we compute the generators, the Hilbert function, and the Hilbert polynomial of the projective closure of affine lines which are parallel to the coordinate axes and pass through a lattice of points. We also consider the Cohen-Macaulay and seminormality property of their homogeneous coordinate ring. These lines are said to form a grid.     

Weighted Projective Lines as Fine Moduli Spaces of Quiver Representations

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their canonical bundles, and show that they are rarely tilting. We also give a moduli construction for these total spaces for weighted projective lines with three orbifold points.     

Smooth threefolds in ${\Bbb P}^5$ without apparent triple or quadruple points and a quadruple-point formula

For a projective variety X of codimension 2 in defined over the complex number field , it is traditionally said that X has no apparent -ple points if the -secant lines of X do not fill up the ambient projective space , equivalently, the locus of -ple points of a generic projection of X to ${\Bbb P}^{n+1}$ is empty. We show that a smooth threefold in has no apparent triple points if and only if it is contained in a quadric hypersurface. We also obtain an enumerative formula counting the quadrisecant lines of X passing through a general point of and give necessary cohomological conditions for smooth threefolds in without apparent quadruple points. This work is intended to generalize the work of F. Severi [fSe] and A. Aure [Au], where it was shown that a smooth surface in has no triple points if and only if it is either a quintic elliptic scroll or contained in a hyperquadric. Furthermore we give open questions along these lines. Received: 24 January 2000 / Published online: 18 June 2001     

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.     

Projective Minkowski set

The Minkowski set or the central symmetry set (CSS) of a smooth curve Γ on the affine plane is the envelope of chords connecting pairs of points such that the tangents to Γ at them are parallel. Singularities of CSS are of interest, in particular, for applications (for example, in computer graphics). A generalization of the Minkowski set is considered in the paper, namely, the projective Minkowski set with respect to a line on the plane; in the case of general position, we describe its singularities and the bifurcation set of lines corresponding to lines defining the projective Minkowski set having singularities being more degenerate than those of the Minkowski set for a generic line.     

Configurations of Flats, I: Manifolds of Points in the Projective Line

The topological manifolds arising from configurations of points in the real and complex projective lines are classified. Their topology and combinatorics are described for the real case. A general setting for the study of the spaces of configurations of flats is established and a projective duality among them is proved in its full generality.     

Existenzbedingungen für Dualitäten projektiver Ebenen

In a projective plane a permutation of the set of all points and lines is constructed, using only the operations join and meet. Under certain conditions (identities in a coordinatizing ternary field; special cases of Desargues- and Pappos-theorem) this permutation is a duality. For a topological projective plane this duality proves, that point space and line space of the plane are homeomorphic.     

Rational points of bounded height on projective surfaces

We give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces.     

Planar sections of the quadric of Lie cycles and their Euclidean interpretations

All cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three dimensional quadric in four dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles. This paper is a careful examination of the interpretation, in terms of systems of cycles in the Euclidean plane, of fundamental incidence configurations involving this quadric in projective space. These interpretations yield new and striking theorems of Euclidean geometry.     

Surfaces of order three with a peak. II

In [1], we presented a theory of surfaces of order three in real projective three-spaceP ³. In the present paper, we prove that a surfaceF of order three with a peak contains one, two or three lines and there are four types of suchF based upon the configuration of these lines. We describe eachF by determining the existence and the distribution of elliptic, parabolic and hyperbolic points; that is, points ofF not lying on any line contained inF.     

A remarkable configuration: A generalization of computergenerated results by S.-C. Chou and its connection with a plane figure by A. Emch

The paper's starting point are four theorems on conics which can be found in a collection of computer proved results by C.-S. Chou from 1987. It not only contains a generalization of two of Chou's results but also a plane figure consisting of points, lines and conics. A suitable notation will reveal a striking symmetry of this figure. Moreover, it turns out that a plane figure from 1940 found by A. Emch using algebraic methods is very similar to ours, which we obtained synthetically. As an application in finite geometry we have gone some way towards regarding our figure as a real projective model of the finite projective plane of order 4.Dedicated to Dr. J. F. Rigby on the occasion of his 65th birthday