On the embedding of some linear spaces in finite projective planes

The problem of embedding of linear spaces in finite projective planes has been examined by several authors ([1], [2], [3], [4], [5], [6]). In particular, it has been proved in [1] that a linear space which is the complement of a projective or affine subplane of order m is embeddable in a unique way in a projective plane of order n. In this article, we give a generalization of this result by embedding linear spaces in a finite projective plane of order n, which are complements of certain regularA-affine linear spaces with respect to a finite projective plane.     

Pasch geometric spaces inducing finite projective planes

Summary A geometric space over a geometric sfield of dimension three induces a protective plane. A relation between the order of the projective plane and that of the geometric sfield is obtained. For a particular order of the sfield, the induced projective plane is shown to be desarguesian.     

4-Webs on hypersurfaces of 4-axial space

V. B. Lazareva investigated 3-webs formed by shadow lines on a surface embedded in 3-dimensional projective space and assumed that the lighting sources are situated on 3 straight lines. The results were used, in particular, for the solution of the Blaschke problem of classification of regular 3-webs formed by pencils of circles in a plane. In the present paper, we consider a 4-web W formed by shadow surfaces on a hypersurface V embedded in 4-dimensional projective space assuming that the lighting sources are situated on 4 straight lines. We call the projective 4-space with 4 fixed straight lines a 4-axial space. Structure equations of 4-axial space and of the surface V , asymptotic tensor of V , torsions and curvatures of 4-web W, and connection form of invariant affine connection associated with 4-web W are found.     

Semistability of Lazarsfeld–Mukai bundles via parabolic structures

Our aim in this article is to produce new examples of semistable Lazarsfeld–Mukai bundles on smooth projective surfaces X using the notion of parabolic vector bundles. In particular, we associate natural parabolic structures to any rank two (dual) Lazarsfeld–Mukai bundle and study the parabolic stability of these parabolic bundles. We also show that the orbifold bundles on Kawamata coverings of X corresponding to the above parabolic bundles are themselves certain (dual) Lazarsfeld–Mukai bundles. This gives semistable Lazarsfeld–Mukai bundles on Kawamata covers of the projective plane and of certain K3 surfaces.     

Small sets of even type and codewords

We examine some geometric configurations of points in designs that give rise to vectors in the codes associated with the designs. In particular we look at small sets of points in projective planes of even order that are met evenly by all the lines of the plane, and find vectors of small weight in the binary hull and in the code's orthogonal. Dedicated to Professor Helmut Karzel on the occasion of his 70^th birthday     

Triangles in Euclidean Arrangements

The number of triangles in arrangements of lines and pseudolines has been the object of some research. Most results, however, concern arrangements in the projective plane. In this article we add results for the number of triangles in Euclidean arrangements of pseudolines. Though the change in the embedding space from projective to Euclidean may seem small there are interesting changes both in the results and in the techniques required for the proofs. In 1926 Levi proved that a nontrivial arrangement—simple or not—of n pseudolines in the projective plane contains at least n triangles. To show the corresponding result for the Euclidean plane, namely, that a simple arrangement of n pseudolines contains at least n-2 triangles, we had to find a completely different proof. On the other hand a nonsimple arrangement of n pseudolines in the Euclidean plane can have as few as 2n/3 triangles and this bound is best possible. We also discuss the maximal possible number of triangles and some extensions. Received February 12, 1998, and in revised form April 7, 1998.     

Harmonic conjugation in harmonic matroids

We study a generalization of the concept of harmonic conjugation from projective geometry and full algebraic matroids to a larger class of matroids called harmonic matroids. We use harmonic conjugation to construct a projective plane of prime order in harmonic matroids without using the axioms of projective geometry. As a particular case we have a combinatorial construction of a projective plane of prime order in full algebraic matroids.     

Immersions of the projective plane with one triple point

We consider C^∞ generic immersions of the projective plane into the 3-sphere. Pinkall has shown that every immersion of the projective plane is homotopic through immersions to Boy's immersion, or its mirror. There is another lesser-known immersion of the projective plane with self-intersection set equivalent to Boy's but whose image is not homeomorphic to Boy's. We show that any C^∞ generic immersion of the projective plane whose self-intersection set in the 3-sphere is connected and has a single triple point is ambiently isotopic to precisely one of these two models, or their mirrors. We further show that any generic immersion of the projective plane with one triple point can be obtained by a sequence of toral and spherical surgical modifications of these models. Finally we present some simple applications of the theorem regarding discrete ambient automorphism groups; image-homology of immersions with one triple point; and almost tight ambient isotopy classes.     

Zur multiplikativen Archimedizität in projektiven ebenen

We determine the local-Archimedean orderings of projective planes over ternary rings whose multiplicative loops of positive elements are Archimedean. In particular, we prove that a projective plane over an Archimedean, linear ternary ring with associative multiplication is Archimedean.     

On Hadamard designs associated with a hyperoval

Hadamard designs which can be associated with a hyperoval of a projective plane of even order are investigated. In particular, when is a translation hyperoval, these designs are shown to contain restrictions that are isomorphic to the 2-design of points and hyperplanes of a projective geometry overGF(2).     

On FC-Purity and I-Purity of Modules and Köthe Rings

In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module.     

The point and line space of a compact projective plane are homeomorphic

The local Hopf bundles at the points and lines of a compact projective plane with manifold lines are all weakly equivalent. In particular, the point space and the line space of such a projective plane are homeomorphic. This is a consequence of the following topological result. Theorem. Let $\xi,\xi'$ be orientable topological $\mathbb{R}^n$ -bundles over an n-dimensional CW-complex. If $\xi$ and $\xi'$ are fibre homotopy equivalent and stably equivalent, then $\xi$ and $\xi'$ are equivalent. Received: 10 December 1998 / Accepted: 18 May 1999     

Subregular Spreads of (2n+1,q)

In this paper, we develop some of the theory of spreads of projective spaces with an eye towards generalizing the results of R. H. Bruck (1969,in“Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514, Univ. of North Carolina Press, Chapel Hill). In particular, we wish to generalize the notion of asubregularspread to the higher dimensional case. Most of the theory here was anticipated by Bruck in later papers; however, he never provided a detailed formulation. We fill this gap here by developing the connections between a regular spread of (2n+1)-dimensional projective space and ann-dimensional circle geometry, which is the appropriate generalization of the Miquelian inversive plane. After developing this theory, we provide a fairly general method for constructing subregular spreads of (5,q). Finally, we explore a special case of this construction, which yields several examples of three-dimensional subregular translation planes which are not André planes.     

Remarks on the Gieseker-Petri divisor in genus eight

In the moduli space of curves of genus 8, M ₈, denote by GP ₈ the locus of curves that do not satisfy the Gieseker-Petri theorem. In this short note we study the projective plane models of curves of genus 8 that do not satisfy the Gieseker-Petri theorem. We use these projective models to exhibit an irreducible divisorial component in GP ₈ and we show that GP ₈ is an irreducible divisor.     

On Planes of Order p2 in Which Every Quadrangle Generates a Subplane of Order p

After Gleason's result, in the late fifties the following conjecture appeared: if in a finite projective plane every quadrangle is contained in a unique Desarguesian proper subplane of order p, then the plane is Desarguesian (and its order is p ^d for some d). In this paper we prove the conjecture in the case when the plane is of order p ² and p is a prime.     

New integer pairs for Hjelmslev planes

Associated with every finite projective Hjelmslev plane is an invariant pair(t, r); t is the order of the Hjelmslev plane andr is the order of the underlying projective plane. The aim of this paper is to give some new constructions of Hjelmslev planes with an invariant pair (t, 2). First we construct a PH-plane with the invariant pair (20, 2). Using this, 16 more invariant pairs (t, 2) witht 1000 are obtained. In all, we thus obtain 17 new PH-planes with invariant pairs (t, 2),t 1000.     

Pfaffian labelings and signs of edge colorings

We relate signs of edge-colorings (as in classical Penrose’s result) with “Pfaffian labelings”, a generalization of Pfaffian orientations, whereby edges are labeled by elements of an Abelian group with an element of order two. In particular, we prove a conjecture of Goddyn that all k-edge-colorings of a k-regular Pfaffian graph G have the same sign. We characterize graphs that admit a Pfaffian labeling in terms of bricks and braces in their matching decomposition and in terms of their drawings in the projective plane. Partially supported by NSF grants 0200595 and 0354742.