Blocking sets in chain geometries
In this paper we give some basic results on blocking sets on minimum size for a finite chain geometry.
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arXiv subjects
Andrea Blunck
Publications and source records attributed to Andrea Blunck.
Lifting of divisible designs
The aim of this paper is to present a construction of $t$-divisible designs for $t>3$, because such divisible designs seem to be missing in the literature. To this end, tools such as finite projective spaces and their algebraic varieties are employed. More precisely, in a first step an abstract construction, called $t$-lifting, is developed. It starts from a set $X$ containing a $t$-divisible design and a group $G$ acting on $X$. Then several explicit examples are given, where $X$ is a subset of $PG(n,q)$ and $G$ is a subgroup of $GL_{n+1}(q)$. In some cases $X$ is obtained from a cone with a Veronesean or an $h$-sphere as its basis. In other examples $X$ arises from a projective embedding of a Witt design. As a result, for any integer $t\geq 2$ infinitely many non-isomorphic $t$-divisible designs are found.
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Divisible designs from twisted dual numbers
The generalized chain geometry over the local ring $K(ε;σ)$ of twisted dual numbers, where $K$ is a finite field, is interpreted as a divisible design obtained from an imprimitive group action. Its combinatorial properties as well as a geometric model in 4-space are investigated.
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Radical parallelism on projective lines and non-linear models of affine spaces
We introduce and investigate an equivalence relation called "radical parallelism" on the projective line over a ring. It is closely related with the Jacobson radical of the underlying ring. As an application, we present a rather general approach to non-linear models of affine spaces and discuss some particular examples.
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Projective Representations II. Generalized chain geometries
In this paper, projective representations of generalized chain geometries are investigated, using the concepts and results of part I. In particular, we study under which conditions such a projective representation maps the chains of a generalized chain geometry $Σ(F,R)$ to reguli; this mainly depends on how the field $F$ is embedded in the ring $R$. Moreover, we determine all bijective morphisms of a certain class of generalized chain geometries with the help of projective representations.
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On distant-isomorphisms of projective lines
We determine all distant-isomorphisms between projective lines over semilocal rings. In particular, for those semisimple rings that do not have a simple component which is isomorphic to a field, every distant isomorphism arises from a Jordan isomorphism of rings and a projectivity. We show this by virtue of a one-one correspondence linking the projective line over a semisimple ring with a Segre product of Grassmann spaces.
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The Dual of a Chain Geometry
We introduce and discuss the dual of a chain geometry. Each chain geometry is canonically isomorphic to its dual. This allows us to show that there are isomorphisms of chain geometries that arise from antiisomorphisms of the underlying rings.
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Jordan homomorphisms and harmonic mappings
We show that each Jordan homomorphism $R\to R'$ of rings gives rise to a harmonic mapping of one connected component of the projective line over $R$ into the projective line over $R'$. If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which is defined on the entire projective line over $R$.
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On bijections that preserve complementarity of subspaces
The set $G$ of all $m$-dimensional subspaces of a $2m$-dimensional vector space $V$ is endowed with two relations, complementarity and adjacency. We consider bijections from $G$ onto $G'$, where $G'$ arises from a $2m'$-dimensional vector space $V'$. If such a bijection $ϕ$ and its inverse leave one of the relations from above invariant, then also the other. In case $m\geq 2$ this yields that $ϕ$ is induced by a semilinear bijection from $V$ or from the dual space of $V$ onto $V'$. As far as possible, we include also the infinite-dimensional case into our considerations.
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The Connected Components of the Projective Line over a Ring
The main result of the present paper is that the projective line over a ring $R$ is connected with respect to the relation "distant" if, and only if, $R$ is a $GE_2$-ring.
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Projective Representations I. Projective lines over rings
We discuss representations of the projective line over a ring $R$ with 1 in a projective space over some (not necessarily commutative) field $K$. Such a representation is based upon a $(K,R)$-bimodule $U$. The points of the projective line over $R$ are represented by certain subspaces of the projective space $P(K,U\times U)$ that are isomorphic to one of their complements. In particular, distant points go over to complementary subspaces, but in certain cases, also non-distant points may have complementary images.
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Affine Spaces within Projective Spaces
We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finite-dimensional or pappian.
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Extending the Concept of Chain Geometry
We introduce the chain geometry $Σ(K,R)$ over a ring $R$ with a distinguished subfield $K$, thus extending the usual concept where $R$ has to be an algebra over $K$. A chain is uniquely determined by three of its points, if, and only if, the multiplicative group of $K$ is normal in the group of units of $R$. This condition is not equivalent to $R$ being a $K$-algebra. The chains through a fixed point fall into compatibility classes which allow to describe the residue at a point in terms of a family of affine spaces with a common set of points.
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Projective lines over Jordan systems and geometry of Hermitian matrices
Any set of $σ$-Hermitian matrices of size $n \times n$ over a field with involution $σ$ gives rise to a projective line in the sense of ring geometry and a projective space in the sense of matrix geometry. It is shown that the two concepts are based upon the same set of points, up to some notational differences.
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Geometric structures on finite- and infinite-dimensional Grassmannians
In this paper, we study the Grassmannian of n-dimensional subspaces of a 2n-dimensional vector space and its infinite-dimensional analogues. Such a Grassmannian can be endowed with two binary relations (adjacent and distant), with pencils (lines of the Grassmann space) and with so-called Z-reguli. We analyse the interdependencies among these different structures.
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Invertible Symmetric 3 x 3 Binary Matrices and GQ(2,4)
We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3 x 3 matrices over GF(2) and the points of the generalized quadrangle GQ(2,4). The 15 matrices with eigenvalue one correspond to a copy of the subquadrangle GQ(2,2), whereas the 12 matrices without eigenvalues have their geometric counterpart in the associated double-six. The fine details of this correspondence, including the precise algebraic meaning/analogue of collinearity, are furnished by employing the representation of GQ(2,4) as a quadric in PG(5,2) of projective index one. An interesting physical application of our findings is also mentioned.
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