Embedding of circulant graphs and generalized Petersen graphs on projective plane

Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane are described, the number of embeddings of C(2n + 1; {1, n}) on the projective plane follows, then the number of embeddings of the generalized Petersen graph P(2n +1, n) on the projective plane is deduced from that of C(2n +1; {1, n}), because C(2n + 1;{1, n}) is a minor of P(2n + 1, n), their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P(2n, 2) on the projective plane is also obtained.     

Minimal topological projective planes

The atoms of the lattice of compatible topologies on a given projective plane are examined. The notion of a field of type V is generalized to ternary fields of type V. These are always minimal, and they arise for example as coordinate structure of strictly uniformizable or of orderable topological planes. Hence, such planes are minimal.Dedicated to Professor Helmut Karzel on his 60^th birthday     

Cartesian groups not belonging to topological projective planes

A new class of topological Cartesian groups is introduced by bending the lines of the affine coordinate plane over an ordered skewfield countably infinitely often. These Cartesian groups belong to topological affine planes with continuous parallelism, but they need not yield a topological projective plane if the topology is not equal to the order topology of the skewfield, in contrast to the results of HARTMANN [6] for generalized Moulton planes. Dedicated to Professor Dr. Gerhard Grimeisen on his 70 ^th birthday     

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.     

Some subsets of points in the plane associated to truncated Reed-Muller codes with good parameters

We gives some examples of subsets of points in the projective plane associated to truncated generalized projective Reed-Muller codes with good parameters, of dimensions 6 and 10 over GF(7), GF(8) and GF(9).     

A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

若干单群与射影平面直射群

设G是有限群,H(?)G。如果H≌~2B_2(q)或H≌~2G_2(q)或H≌PSU(3,q),则G不与任何射影平面的点传递直射群同均。本文对以下问题给出了一般方法:证明以某些几乎单群为点传递自同构群的线性空间不是射影平面。     

Ten Exceptional Geometries from Trivalent Distance Regular Graphs

The ten distance regular graphs of valency 3 and girth > 4 define ten non-isomorphic neighborhood geometries, amongst which a projective plane, a generalized quadrangle, two generalized hexagons, the tilde geometry, the Desargues configuration and the Pappus configuration. All these geometries are bislim, i.e., they have three points on each line and three lines through each point. We study properties of these geometries such as embedding rank, generating rank, representation in real spaces, alternative constructions. Our main result is a general construction method for homogeneous embeddings of flag transitive self-polar bislim geometries in real projective space.     

Degeneration of plane affine Stolyarov connections

Consider a distribution of a plane in a projective space. A way of defining a plane affine Stolyarov connection associated with this distribution is proposed. It is set by the field of a connection object consisting of a connection quasitensor and a linear connection object. The object of this generalized affine connection defines torsion and curvature objects. We show that these objects are tensors. Conditions under which a plane affine Stolyarov connection is torsion-free or curvature-free are described. It is proved that the generalized affine connection with the connection quasitensor is the generalized Kronecker symbol degenerated into a linear connection.     

Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P².The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E⊂P². In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P²?E.     

Exceptional presentations of three generalized hexagons of order 2

The generalized hexagons associated with L₃(2), U₃(3), ³D₄(2), respectively, are presented as subconfigurations of the projective plane over the complex, quaternionic, octonionic field, respectively. This leads to an embedding of Aut ³D₄(2) in the Lie group of type F₄(R).     

The Classification of Projective Planes of Order q3 with Cone-Representations in PG (6, q)

The classification of cone-representations of projective planes of orderq ³ of index 3 and rank 4 (and so in PG(6,^q)) is completed. Any projective plane with a non-spread representation (being a cone-representation of the second kind) is a dual generalised Desarguesian translation plane, as found by Jha and Johnson, and conversely. Indeed, given any collineation of PG(2,^q) with no fixed points, there exists such a projective plane of order q³ , where q is a prime power, that has the second kind of cone-representation of index 3 and rank 4 in PG(6,q). An associated semifield plane of order q ³ is also constructed at most points of the plane. Although Jha and Johnson found this plane before, here we can show directly the geometrical connection between these two kinds of planes.     

On surfaces and their contours

We prove two theorems concerning the global behaviour of a smooth compact surfaceS, without boundary, embedded in a real projective space or mapped to a plane. Our starting point is an analysis of the orientability properties of the normal bundle of a singular projective curve. Then we see how an excellent projection fromS to the Euclidean plane gives rise to integral relations linking the singularities of the apparent contour. Finally, given an embedding ofS in RPⁿ, we look at the discriminant Δ^* of a net of hyperplanes that intersectsS in a generic way, obtaining a characterization of Δ^* in terms of mod.2 cohomology invariants.     

On representing sylvester-gallai designs

A misstated conjecture in [3] leads to an interesting (1, 3) representation of the 7-point projective plane inR ⁴ where points are represented by lines and planes by 3-spaces. The corrected form of the original conjecture will be negated if there is a (1, 3) representation of the 13-point projective plane inR ⁴ but that matter is not settled.     

Geometry of free cyclic submodules over ternions

Given the algebra T of ternions (upper triangular 2×2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T ². This set of points can be represented as a set of planes in the projective space over F ⁶. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F-linear antiautomorphism, the plane model of our projective line does not admit any duality.     

Realizability of the torus and the projective plane in ℝ4

We show that every triangulation of the projective plane or the torus is isomorphic to a subcomplex of the boundary complex of a simplicial 5-dimensional convex polytope and thus linearly embeddable in ℝ⁴.     

Cyclic projective planes and binary,extended cyclic self-dual codes

If P is a cyclic projective plane of order n, we give number theoretic conditions on n² + n + 1 so that the binary code of P is contained in a binary cyclic code C whose extension is self-dual. When this containment occurs C does not contain any ovals of P. As a corollary to these conditions we obtain that the extended binary code of a cyclic projective plane of order 2^s is contained in a binary extended cyclic self-dual code if and only if s is odd.     

On finite sets of points inP 3

Given a finite set of points Γ₀ which span a projective spaceP ³, we show here that a plane spanned by points of Γ₀ can be a neighbour of at most eight points of Γ₀, these being the vertices of a projective cube; the common neighbour plane is then elementary with the three only points of Γ₀ in it being diagonal points of the cube. This extends toP ³ some results of L. M. Kelly and W. O. J. Moser in the planeP ².     

A lower bound for the minimum weight of the dual 7-ary code of a projective plane of order 49

Existing bounds on the minimum weight d ^⊥ of the dual 7-ary code of a projective plane of order 49 show that this must be in the range 76 ≤ d ^⊥ ≤ 98. We use combinatorial arguments to improve this range to 88 ≤ d ^⊥ ≤ 98, noting that the upper bound can be taken to be 91 if the plane has a Baer subplane, as in the desarguesian case. A brief survey of known results for the minimum weight of the dual codes of finite projective planes is also included. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

Integral relations for pointed curves in a real projective plane

We study some global projective properties of real plane curves with cusps, beaks and normal crossings. Starting from Fabricius-Bjerre's formula on the singularities of a curve in an affine plane, we describe its extension to a projective setting. Given a curve RP², by fixing a base point we associate some indices to its singularities and double tangents. We prove two global relations linking these entities together.