Regular Spreads Research Articles

Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case

Abstract Using the Klein correspondence, regular parallelisms of PG(3, ℝ) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining (hfd) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here we refine our simplified approach in order to obtain similar results for regular parallelisms of oriented lines. As a consequence, we can demonstrate that for oriented parallelisms, as we call them, there are distinctly more possibilities than in the non-oriented case. The proofs require a thorough analysis of orientation in projective spaces (as manifolds and as lattices) and in projective planes and, in particular, in translation planes. This is used in order to handle continuous families of oriented regular spreads in terms of the Klein model of PG(3, ℝ). This turns out to be quite subtle. Even the definition of suitable classes of dual objects modeling oriented parallelisms is not so obvious.

Pencilled regular parallelisms

Over any field \({\mathbb{K}}\), there is a bijection between regular spreads of the projective space \({{\rm PG}(3,\mathbb{K})}\) and 0-secant lines of the Klein quadric in \({{\rm PG}(5,\mathbb{K})}\). Under this bijection, regular parallelisms of \({{\rm PG}(3,\mathbb{K})}\) correspond to hyperflock determining line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is defined to be pencilled if it is composed of pencils of lines. We present a construction of pencilled hfd line sets, which is then shown to determine all such sets. Based on these results, we describe the corresponding regular parallelisms. These are also termed as being pencilled. Any Clifford parallelism is regular and pencilled. From this, we derive necessary and sufficient algebraic conditions for the existence of pencilled hfd line sets.

Clifford parallelism: old and new definitions, and their use

Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space $${{\rm PG}(3,\mathbb{R})}$$ whose (complex) focal lines (=directrices) form a regulus contained in an imaginary quadric (D1 = Klein’s definition). Our new access to the topic ‘Clifford parallelism’ is free of complexification and involves Klein’s correspondence λ of line geometry together with a bijective map γ from all regular spreads of $${{\rm PG}(3,\mathbb{R})}$$ onto those lines of $${{\rm PG}(5,\mathbb{R})}$$ having no common point with the Klein quadric; a regular parallelism P of $${{\rm PG}(3,\mathbb{R})}$$ is Clifford, if the spreads of P are mapped by γ onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with γ is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of $${{\rm PG}(3,\mathbb{R})}$$ are Clifford (D3 = dimensionality definition). Submission of (D2) to λ−1 yields a complexification free definition of a Clifford parallelism which uses only elements of $${{\rm PG}(3,\mathbb{R})}$$ : A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simultaneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group $${Aut_e({\bf P}_{\bf C})}$$ of all automorphic collineations and dualities of the Clifford parallelism P C and show $${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$$ .

The Good - or the Bad and the Ugly? Corporate Governance and Risk Taking in Banks

This paper analyzes the influence a bank’s corporate governance structure has on its risk taking. To do so, the paper specifically looks at illiquidity and insolvency risk, as measured by a bank’s liquidity creation and its margin between interest income from loans and interest expenses on deposits. The relevance of this question is rooted in banks’ core business models: maturity transformation. By transforming short-term liabilities into long-term assets banks generate profits through the exploitation of the yield curve spread. However, in times of tightening or inverting yield curve spreads the banks’ transformation margins narrow. Management then faces a dilemma: accepting narrowed spreads and lower profits or increasing the volume of maturity transformation. Either option poses a threat to the bank, the former of insolvency, the latter of illiquidity. This paper tries to answer the question of how banks with different owner and manager structures deal with this trade-off. Analyzing a sample of 259 U.S. commercial banks over the period 1998-2010, the paper finds that banks with low separation of ownership and control increase their liquidity risk in times of narrowing or inverted yield curve structures to keep profit levels up through volume effects. In contrast, banks with higher separation of ownership and control opt for less liquidity risk, also in times of regular yield curve spreads, and thereby also accept lower interest rate margins.

Parallelisms of $${{\rm PG}(3, {\mathbb R})}$$ composed of non-regular spreads

Any continuous strictly monotonic function $${F : {{\mathbb R}^{{\geq}0} \to {\mathbb R}}}$$ with F(0) = 0 and F(t) → ∞ for t → ∞ gives rise to a topological rotational spread of $${{\rm PG}\,(3,{\mathbb R})}$$ ; this spread is non-regular, if F is not linear. The action of the group $${SO_3({\mathbb R})}$$ on this spread yields a topological parallelism of $${{\rm PG}\,(3,{\mathbb R})}$$ . The article also contains a short investigation on rotational spreads. Moreover, we construct a parallelism P 72 of $${{\rm PG}\,(3,{\mathbb R})}$$ which is composed of piecewise regular spreads each consisting of two segments which are tacked together along a common regulus. Using Klein’s correspondence of line geometry and the Thas–Walker construction we represent every parallel class of P 72 via two parallel half-lines being non-interior to a given sphere in $${{\mathbb R}^3}$$ . The parallelism P 72 contains exactly one regular spread, all other members of P 72 are piecewise regular spreads with two segments. However, P 72 is not topological.

On polyvectors of vector spaces and hyperplanes of projective Grassmannians

We investigate relationships between polyvectors of a vector space V, alternating multilinear forms on V, hyperplanes of projective Grassmannians and regular spreads of projective spaces. Suppose V is an n-dimensional vector space over a field F and that A n - 1 , k ( F ) is the Grassmannian of the ( k − 1)-dimensional subspaces of PG( V) (1 ⩽ k ⩽ n − 1). With each hyperplane H of A n - 1 , k ( F ) , we associate an ( n − k)-vector of V (i.e., a vector of ∧ n− k V) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of A n - 1 , k ( F ) , i.e., how isomorphism of hyperplanes can be recognized in terms of their representative vectors. Special attention is paid here to the case n = 2 k and to those isomorphisms which arise from dualities of PG( V). We also prove that with each regular spread of the projective space PG ( 2 k - 1 , F ) , there is associated some class of isomorphic hyperplanes of the Grassmannian A 2 k - 1 , k ( F ) , and we study some properties of these hyperplanes. The above investigations allow us to obtain a new proof for the classification, up to equivalence, of the trivectors of a 6-dimensional vector space over an arbitrary field F , and to obtain a classification, up to isomorphism, of all hyperplanes of A 5 , 3 ( F ) .

Some subspaces of the projective space PG(Λ^kV) related to regular spreads of PG(V)

Let V be a 2m-dimensional vector space over a field F (m ≥ 2) and let k ∈ {1, .. .,2m − 1}. Let A2m−1,k denote the Grassmannian of the (k − 1)-dimensional subspaces of PG(V ) and let egr denote the Grassmann embedding of A2m−1,k into PG(Vk V ). Let S be a regular spread of PG(V ) and let XS denote the set of all (k − 1)-dimensional subspaces of PG(V ) which contain at least one line of S. Then we show that there exists a subspace Σ of PG(Vk V ) for which the following holds: (1) the projective dimension of Σ is equal to 2m k
− 2 · m k
− 1; (2) a (k−1)-dimensional subspace α of PG(V ) belongs to XS if and only if egr(α) ∈ Σ; (3) Σ is generated by all points egr(p), where p is some point of XS.

Hyperflock determining line sets and totally regular parallelisms of PG $${(3,\,{\mathbb{R}})}$$

By a totally regular parallelism of the real projective 3-space \({\Pi_3:={{\rm PG}}(3, \mathbb {R})}\) we mean a family T of regular spreads such that each line of Π3 is contained in exactly one spread of T. For the investigation of totally regular parallelisms the authors mainly employ Klein’s correspondence λ of line geometry and the polarity π5 associated with the Klein quadric H5 (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H5, i.e., a family H of elliptic subquadrics of H5 such that each point of H5 is on exactly one subquadric of H. Moreover, \({\{\pi_5({{\rm span}} \,\lambda(\mathcal {X}))\vert\mathcal {X}\in\bf{T}\}=:\mathcal {H}_{\bf{T}}}\) is a hyperflock determining line set, i.e., a set \({\mathcal {Z}}\) of 0-secants of H5 such that each tangential hyperplane of H5 contains exactly one line of \({\mathcal {Z}}\) . We say that \({{{\rm dim}}({{\rm span}}\,\mathcal {H}_{\bf{T}})=:d_{\bf{T}}}\) is the dimension of T and that T is a dT- parallelism. Clifford parallelisms and 2-parallelisms coincide. The examples of non-Clifford parallelisms exhibited in Betten and Riesinger [Result Math 47:226–241, 2004; Adv Geom 8:11–32, 2008; J Geom (to appear)] are totally regular and of dimension 3. If \({\mathcal{G}}\) is a hyperflock determining line set, then \({\{\lambda^{-1}\,{\rm (}\pi_5(X){\,\cap H_5)\,|\, X\in\mathcal{G}\}}}\) is a totally regular parallelism. In the present paper the authors construct examples of topological (see Definition 1.1) 4- and 5-parallelisms via hyperflock determining line sets.

Partitions in finite geometry and related constant composition codes

We look at the construction of constant composition codes (CCCs) from various types of partitions in finite projective spaces. In particular, we construct robust classes of codes using regular spreads of PG(2n−1,q) and Baer subgeometry partitions of PG(2n,q2). For each class of codes, we bound the minimum distance by considering how such partitions can intersect. As such, we prove results about the intersection of regular spreads and Baer subgeometry partitions, two of the classical partitions generated by subgroups of a Singer group. In addition, we examine other partitions of objects embedded in finite projective spaces and their associated codes. In each case, we compare our codes to a code of comparable parameters that meets the Plotkin bound.

Hermitian indicator sets

Abstract Indicator sets arising from symplectic line-spreads of a 3-dimensional projective space are characterized. Moreover, locally Hermitian ovoids of a Hermitian surface (3, q 2) are studied in terms of “non-equivalent” associated indicator sets. Translation locally Hermitian ovoids of (3, q 2) arising from regular spreads of PG(3, q), Kantor semifield spreads of PG(3, q) and semifield spreads of PG(3, q) with center are also investigated.

Topological parallelisms of the real projective 3-space

A parallelism of a projective 3-space Π is a family P of spreads such that each line of Π is contained in exactly one spread of P. A parallelism is said to be totally regular, if all its members are regular spreads. By a generalized line star with respect to an elliptic quadric Q of a classical projective 3-space we understand a set \(\cal A\) of 2-secants of Q such that each non-interior point of Q is incident with exactly one line of \(\cal A\). From each generalized line star we can construct a totally regular parallelism which we do in essential by the Thas-Walker construction. A parallelisms of the real projective 3-space PG(3, ℝ) is called topological, if the operation of drawing a line parallel to a given line through a given point is continuous. Clifford parallelisms are topological. Using generalized line stars we exhibit examples of non-Clifford topological parallelisms and of non-topological parallelisms.

Classical mixed partitions

Through a method given in Hirschfeld and Thas (General Galois Geometries, Oxford University Press, Oxford, 1991), a mixed partition of PG(2 n −1, q 2 ) can be used to construct a (2 n −1)-spread of PG(4 n −1, q ) and, hence, a translation plane of order q 2 n . A mixed partition in this case is a partition of the points of PG(2 n −1, q 2 ) into PG( n −1, q 2 )'s and PG(2 n −1, q )'s which we call Baer subspaces. In this paper, we completely classify the mixed partitions which generate regular spreads and, hence, can be classified as classical .

Classical mixed partitions

Through a method given in Hirschfeld and Thas (General Galois Geometries, Oxford University Press, Oxford, 1991), a mixed partition of PG(2 n−1, q 2) can be used to construct a (2 n−1)-spread of PG(4 n−1, q) and, hence, a translation plane of order q 2 n . A mixed partition in this case is a partition of the points of PG(2 n−1, q 2) into PG( n−1, q 2)'s and PG(2 n−1, q)'s which we call Baer subspaces. In this paper, we completely classify the mixed partitions which generate regular spreads and, hence, can be classified as classical.

Piecewise regular spreads

A spread % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $\cal S$ of the real projective 3-space PG(3,ℝ) is called piecewise regular, if, roughly speaking, the Klein image of % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $\cal S$ is composed of two elliptic caps and z elliptic zones (z ∈ {0,1,2,…}); we say that % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $\cal S$ is of segment number z + 2. We use piecewise regular spreads in order to give explicit examples of rigid and hyperrigid spreads. A spread % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $\cal T$ of a projective 3-space II is called rigid, if the only collineation of II leaving % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $\cal T$ invariant is the identity. A rigid spread ℌ is said to be hyperrigid, if there is no duality of II leaving ℌ invariant. We exhibit a 3-parameter family S″ of rigid piecewise regular spreads of segment number 4 and show that S″ contains spreads which represent non-isomorphic rigid 4-dimensional translation planes. Finally, we construct a 7-parameter family H of explicit examples of hyperrigid piecewise regular spreads of segment number 5. In H there are at least four spreads which represent mutually non-isomorphic rigid 4-dimensional translation planes.

On Hermitian Spreads

Let $\perp$ be the polarity of $PG(5,q)$ defined by the elliptic quadric $Q^-(5,q)$. A locally Hermitian spread ${\cal S}$ of $Q^-(5,q)$, with respect to a line $L$, is associated in a canonical way with a spread ${\cal S}_{\Lambda}$ of the $3$-dimensional projective space $L^{\perp} = \Lambda$, and conversely. In this paper we give a geometric characterization of the regular spreads of $\Lambda$ which induce Hermitian spreads of $Q^-(5,q)$.

Regular spreads and chain geometries

Regular spreads and chain geometries

Regular Packings of PG (3,q)

Two regular packings of PG(3,q) are constructed wheneverq≡2(mod3), with each packing admitting a cyclic group of orderq2+q+1 acting regularly on the regular spreads in the packing. The resulting families of translation planes of orderq4include the Lorimer–Rahilly and Johnson–Walker planes of order 16.

Partitioning projective geometries into Segre varieties

Using suitable subgroups of Singer cyclic groups we prove some properties of regular spreads and Segre varieties, which in turn yield a necessary and sufficient condition for partitioning a finite projective space into such varieties.

Bruck’s vision of regular spreads or what is the use of a baer superspace?

Bruck’s vision of regular spreads or what is the use of a baer superspace?

Geradenmodelle der M�bius- und Burau-Geometrien

Let (L, K) be an arbitrary (not necessary abelian) field extension, let V be a (left) vector space over L (and therefore over the subfield K) and let μ be the set of all sub-spaces of the projective space II(V, K). Then the canonical map ϰ: P ≔ V*/K* → U given by ϰ: K*a → L*a/K* ≔ {K*(λa)∣λ ∈ L*{ induces a spreadS in Π(V, K). Furthermore ϰ induces a chain structure by applying ϰ on certain subspaces of U. Our purpose is to give a synthetic description of these chain structures. For a regular SpreadS, i.e. K is commutative, the chain structure is a generalization of the Burau geometries [11]. In this paper we represent the Staudt chains by Segre manifolds.