Automorphisms of Some Topological Regular Parallelisms of {{\rm PG}(3, \mathbb{R})}

In a precedent article we constructed various topological regular parallelisms of the real projective 3-space \({{\rm PG}(3, \mathbb{R})}\) via hyperflock determining line sets of \({{\rm PG}(5, \mathbb{R})}\) (see Betten and Riesinger in Mh Math 161:43–58, 2010). In the present paper we discuss for some of these parallelisms their automorphism groups consisting of all automorphic collineations and all automorphic dualities, especially we compute their group dimension. Thus we are able to present: (1) topological regular 5-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 0, (2) topological regular 4-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 0 or 1, (3) topological regular 3-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 1.     

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 Π ₃ 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 π ₅ associated with the Klein quadric H ₅ (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H ₅, i.e., a family H of elliptic subquadrics of H ₅ such that each point of H ₅ 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 H ₅ such that each tangential hyperplane of H ₅ 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 d _T - 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.     

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 ?? of line geometry together with a bijective map ?? 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 ?? onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with ?? 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 ??^?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 ${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$ .     

Maximal partial line spreads of non-singular quadrics

For $n \ge 9$ , we construct maximal partial line spreads for non-singular quadrics of $PG(n,q)$ for every size between approximately $(cn+d)(q^{n-3}+q^{n-5})\log {2q}$ and $q^{n-2}$ , for some small constants $c$ and $d$ . These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by Gács and Sz?nyi. These results also extend spectrum results on maximal partial line spreads in the finite generalized quadrangles $W_3(q)$ and $Q(4,q)$ by Pepe, Rößing and Storme.     

Piecewise regular spreads

A spread $\cal S$ of the real projective 3-space PG(3,?) is called piecewise regular, if, roughly speaking, the Klein image of $\cal S$ is composed of two elliptic caps and z elliptic zones (z ∈ {0,1,2,…}); we say that $\cal S$ is of segment number z + 2. We use piecewise regular spreads in order to give explicit examples of rigid and hyperrigid spreads. A spread $\cal T$ of a projective 3-space II is called rigid, if the only collineation of II leaving $\cal T$ invariant is the identity. A rigid spread ? is said to be hyperrigid, if there is no duality of II leaving ? 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.     

Severi inequality for varieties of maximal Albanese dimension

In this paper, we prove the general Severi inequality for varieties of maximal Albanese dimension. Suppose that \(X\) is an \(n\) -dimensional projective, normal, minimal and \(\mathbb {Q}\) -Gorenstein variety of general type in characteristic zero. If \(X\) is of maximal Albanese dimension, then \(K^n_X \ge 2 n! \chi (\omega _X)\) .     

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert W*-module

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module l2 A, and give a one to one correspondence between the set of positive self-adjoint regular module operators on l2A and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup {Tt }t∈R + L(l2A) is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L(l2A) is the von Neumann algebra of all adjointable module maps on l2A.     

On the field of definition of p-torsion points on elliptic curves over the rationals

Let $S_\mathbb Q (d)$ be the set of primes $p$ for which there exists a number field $K$ of degree $\le d$ and an elliptic curve $E/\mathbb Q $ , such that the order of the torsion subgroup of $E(K)$ is divisible by $p$ . In this article we give bounds for the primes in the set $S_\mathbb Q (d)$ . In particular, we show that, if $p\ge 11$ , $p\ne 13,37$ , and $p\in S_\mathbb Q (d)$ , then $p\le 2d+1$ . Moreover, we determine $S_\mathbb Q (d)$ for all $d\le 42$ , and give a conjectural formula for all $d\ge 1$ . If Serre’s uniformity problem is answered positively, then our conjectural formula is valid for all sufficiently large $d$ . Under further assumptions on the non-cuspidal points on modular curves that parametrize those $j$ -invariants associated to Cartan subgroups, the formula is valid for all $d\ge 1$ .     

7c-Gorenstein Projective, ;Cc-Gorenstein Injectiveand Wcr-Gorenstei丨丨 Flat Modules

The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.     

A refined stable restriction theorem for vector bundles on quadric threefolds

Let \(E\) be a stable rank 2 vector bundle on a smooth quadric threefold \(Q\) in the projective 4-space \(P\) . We show that the hyperplanes \(H\) in \(P\) for which the restriction of \(E\) to the hyperplane section of \(Q\) by \(H\) is not stable form, in general, a closed subset of codimension at least 2 of the dual projective 4-space, and we explicitly describe the bundles \(E\) which do not enjoy this property. This refines a restriction theorem of Ein and Sols (Nagoya Math J 96:11–22, 1984) in the same way the main result of Coand? (J Reine Angew Math 428:97–110, 1992) refines the restriction theorem of Barth (Math Ann 226:125–150, 1977).     

The Euler class group of a polynomial algebra with coefficients in a line bundle

Let $A$ be a commutative Noetherian ring and $P$ be a projective $A$ -module of rank $=(\text {dim}(A)-1)$ . An intriguing open question is to find the precise obstruction for $P$ to split as: $P\simeq Q\oplus A$ for some $A$ -module $Q$ . In this paper we settle this question when $A=R[T]$ for some ring $R$ containing the field of rationals and $P$ is a projective $A$ -module of rank $=\text {dim}(R)$ .     

The ring of evenly weighted points on the line

Let \(M_w = ({\mathbb {P}}^1)^n /\!/\hbox {SL}_2\) denote the geometric invariant theory quotient of \(({\mathbb {P}}^1)^n\) by the diagonal action of \(\hbox {SL}_2\) using the line bundle \(\mathcal {O}(w_1,w_2,\ldots ,w_n)\) on \(({\mathbb {P}}^1)^n\) . Let \(R_w\) be the coordinate ring of \(M_w\) . We give a closed formula for the Hilbert function of \(R_w\) , which allows us to compute the degree of \(M_w\) . The graded parts of \(R_w\) are certain Kostka numbers, so this Hilbert function computes stretched Kostka numbers. If all the weights \(w_i\) are even, we find a presentation of \(R_w\) so that the ideal \(I_w\) of this presentation has a quadratic Gröbner basis. In particular, \(R_w\) is Koszul. We obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of \(M_w\) .     

Almost Split Sequences and Approximations

Let $\mathcal A$ be an exact category, that is, an extension-closed full subcategory of an abelian category. First, we give new characterizations of an almost split sequence in $\mathcal{A}$ , which yields some necessary and sufficient conditions for $\mathcal A$ to have almost split sequences. Then, we study when an almost split sequence in $\mathcal A$ induces an almost split sequence in an exact subcategory $\mathcal C$ of $\mathcal A$ . In case $\mathcal A$ has almost split sequences and $\mathcal C$ is Ext-finite and Krull–Schmidt, we obtain a necessary and sufficient condition for $\mathcal C$ to have almost split sequences. Finally, we show some applications of these results.     

Relatively congruence-free regular semigroups

Yu, Wang, Wu and Ye call a semigroup \(S\) \(\tau \) -congruence-free, where \(\tau \) is an equivalence relation on \(S\) , if any congruence \(\rho \) on \(S\) is either disjoint from \(\tau \) or contains \(\tau \) . A congruence-free semigroup is then just an \(\omega \) -congruence-free semigroup, where \(\omega \) is the universal relation. They determined the completely regular semigroups that are \(\tau \) -congruence-free with respect to each of the Green’s relations. The goal of this paper is to extend their results to all regular semigroups. Such a semigroup is \(\mathrel {\mathcal {J}}\) -congruence-free if and only if it is either a semilattice or has a single nontrivial \(\mathrel {\mathcal {J}}\) -class, \(J\) , say, and either \(J\) is a subsemigroup, in which case it is congruence-free, or otherwise its principal factor is congruence-free. Given the current knowledge of congruence-free regular semigroups, this result is probably best possible. When specialized to completely semisimple semigroups, however, a complete answer is obtained, one that specializes to that of Yu et al. A similar outcome is obtained for \(\mathrel {\mathcal {L}}\) and \(\mathrel {\mathcal {R}}\) . In the case of \(\mathrel {\mathcal {H}}\) , only the completely semisimple case is fully resolved, again specializing to those of Yu et al.     

Some graftings of complex projective structures with Schottky holonomy

Let ${\mathcal{G} ^{*}(S, \rho)}$ be the graph whose vertices are marked complex projective structures with holonomy ${\rho}$ and whose edges are graftings from one vertex to another. If ${\rho}$ is quasi-Fuchsian, a theorem of Goldman implies that ${\mathcal{G} ^{*}(S, \rho)}$ is connected. If ${\rho ( \pi _{1}(S))}$ is a Schottky group Baba has shown that ${\mathcal{G}(S, \rho)}$ (the corresponding graph for unmarked structures) is connected. For the case that ${\rho ( \pi _{1}(S))}$ is a Schottky group, this paper provides formulae for the composition of graftings in a basic setting. Using these formulae, one can construct an infinite number of (standard) projective structures which can be grafted to a common structure. Furthermore, one can construct pairs of projective structures which can be connected by grafting in an infinite number of ways.     

Pairs of Projections: Geodesics,Fredholm and Compact Pairs

A pair \((P, Q)\) of orthogonal projections in a Hilbert space \( \mathcal{H} \) is called a Fredholm pair if $$\begin{aligned} QP : R(P) \rightarrow R(Q) \end{aligned}$$ is a Fredholm operator. Let \( \mathcal{F} \) be the set of all Fredholm pairs. A pair is called compact if \(P-Q\) is compact. Let \( \mathcal{C} \) be the set of all compact pairs. Clearly \( \mathcal{C} \subset \mathcal{F} \) properly. In this paper it is shown that both sets are differentiable manifolds, whose connected components are parametrized by the Fredholm index. In the process, pairs \(P, Q\) that can be joined by a geodesic (or equivalently, a minimal geodesic) of the Grassmannian of \( \mathcal{H} \) are characterized: this happens if and only if $$\begin{aligned} \dim (R(P)\cap N(Q))=\dim (R(Q)\cap N(P)). \end{aligned}$$      

A Clifford Algebraic Approach to Line Geometry

In this paper we combine methods from projective geometry, Klein’s model, and Clifford algebra. We develop a Clifford algebra whose Pin group is a double cover of the group of regular projective transformations. The Clifford algebra we use is constructed as homogeneous model for the five-dimensional real projective space \({\mathbb {P}^5 (\mathbb{R})}\) where Klein’s quadric \({M^4_2}\) defines the quadratic form. We discuss all entities that can be represented naturally in this homogeneous Clifford algebra model. Projective automorphisms of Klein’s quadric induce projective transformations of \({\mathbb {P}^3 (\mathbb{R})}\) and vice versa. Cayley-Klein geometries can be represented by Clifford algebras, where the group of Cayley-Klein isometries is given by the Pin group of the corresponding Clifford algebra. Therefore, we examine the versor group and study the correspondence between versors and regular projective transformations represented as 4 × 4 matrices. Furthermore, we give methods to compute a versor corresponding to a given projective transformation.     

Stationary and invariant densities and disintegration kernels

Let \(G\) be a locally compact topological group, acting measurably on some Borel spaces \(S\) and \(T\) , and consider some jointly stationary random measures \(\xi \) on \(S\times T\) and \(\eta \) on \(S\) such that \(\xi (\cdot \times T)\ll \eta \) a.s. Then there exists a stationary random kernel \(\zeta \) from \(S\) to \(T\) such that \(\xi =\eta \otimes \zeta \) a.s. This follows from the existence of an invariant kernel \(\varphi \) from \(S\times {\mathcal {M}}_{S\times T}\times {\mathcal {M}}_S\) to \(T\) such that \(\mu =\nu \otimes \varphi (\cdot ,\mu ,\nu )\) whenever \(\mu (\cdot \times T)\ll \nu \) . Also included are some related results on stationary integration, absolute continuity, and ergodic decomposition.