Partial lineations between arguesian projective spaces

Abstract

E. ARTIN (1957), R. BAER (1952) and J.DIEUDONNE (1951) emphasized many times the structural identity between classical projective geometry and linear algebra over a division ring. Then Baer pointed out a possible extension of this structural identity to the case of a ring, generating intense research activity in the area of geometric algebra over rings. In this direction the most significant results appear: Ojanguren and Sridharan’s article on the fundamental theorem of projective geometry over a commutative ring; some theorems due to Klingenberg, Bass and Suslin on the structure of the general linear group over appropriate rings and some valuable notes of O’Meara’s on the automorphisms of linear groups over an integral domain.

The fundamental theorem of projective geometry

Let X be a paracompact complex manifold of dimension «J>1. We call X a weakly 1-complete manifold if there exists a C°° plurisubharmonic function (? on X such that for every ceR (real number), <p~1((— oo? c)} is relatively compact in X. cp is called an exhaustion function of X. It is well known that holomorphically convex manifolds are weakly 1-complete. The converse is not true in general (cf. [1]), so one is led to the problem of seeking natural additional conditions which make weakly 1-complete manifolds holomorphically convex. The content of this article is divided into two parts; Section 1 is devoted to prove some properties of weakly 1-complete manifolds which have a nonconstant holomorphic function. In Section 2, first we present an application of the Nakano's vanishing theorem to the Levi problem on projective spaces and hyperquadrics. These results are not new (cf. [3]) but the method will be of some interest. Next, combining the Nakano's vanishing theorem with the result in Section 1 and a well known theorem (due to Bonnet) of differential geometry we obtain the following.

Computer graphics is ostensibly based on projective geometry. The graphics pipeline—the sequence of functions applied to 3D geometric primitives to determine a 2D image—is described in the graphics literature as taking the primitives from Euclidean to projective space, and then back to Euclidean space. This is a weak foundation for computer graphics. An instructor is at a loss: one day entering the classroom and invoking the established and venerable theory of projective geometry while asserting that projective spaces are not separable, and then entering the classroom the following week to tell the students that the standard graphics pipeline performs clipping not in Euclidean, but in projective space—precisely the operation (deciding sidedness, which depends on separability) that was deemed nonsensical. But there is no need to present Blinn and Newell’s algorithm (Comput. Graph. 12, 245–251, 1978; Commun. ACM 17, 32–42, 1974)—the crucial clipping step in the graphics pipeline and, perhaps, the most original knowledge a student learns in a fourth-year computer graphics class—as a clever trick that just works. Jorge Stolfi described in 1991 oriented projective geometry. By declaring the two vectors $(x,y,z,w)^{{\ensuremath {\mathsf {T}}}}$and $(-x,-y,-z,-w)^{{\ensuremath {\mathsf {T}}}}$distinct, Blinn and Newell were already unknowingly working in oriented projective space. This paper presents the graphics pipeline on this stronger foundation.

Chapter 2 - Projective and Affine Geometry over Division Rings

Coherent state embeddings, polar divisors and Cauchy formulas

Linearity and complements in projective space

Let G G be a locally compact group and E E be a G G -space. An irreducible probability measure μ \mu on G G is said to have Liouville property on E E if G G -invariant functions on E E are the only continuous bounded functions on E E that satisfy the mean value property with respect to μ \mu . We first prove that the random walk induced by μ \mu on E E is transient outside a closed set and on the closed set μ \mu has Liouville provided G G is metrizable and E E is compact and metrizable. We mainly consider actions on vector spaces and projective spaces. We show that measures on G L ( V ) GL(V) that are supported inside a ball of radius less than 1 1 have Liouville property on V V . We also prove that measures on G L ( R 2 ) GL(\mathbb {R} ^2) have Liouville property on the projective line. We next exhibit subgroups of G L ( V ) GL(V) so that irreducible measures on such subgroups have Liouville on the projective space P ( V ) \mathbb {P} (V) of V V . We also prove that irreducible measures on S L ( V ) SL(V) have Liouville property on P ( S L ( V ) ) \mathbb {P} (\mathcal {SL} (V)) where S L ( V ) \mathcal {SL} (V) is the Lie algebra of S L ( V ) SL(V) .

We survey, following Fulton, the connectedness principle of Enriques–Severi–Zariski claiming that limits of irreducible varieties remain connected. Some deep generalizations are reported following the circle of ideas which led to the Connectedness Theorem of Fulton–Hansen and to some new results related to it. We include a proof of the Fulton–Hansen Theorem according to Deligne and we describe its consequences for the geometry of embedded projective varieties proved by Zak: Theorem on Tangency, finiteness of the Gauss map, dimension of the dual variety, hyperplane sections of low-codimensional varieties. These are modern and deep generalizations of many classical theorems in projective geometry which usually dealt with general objects. We also define some classical tangential invariants of projective varieties and connect them to tangential projections and to Scorza’s Lemma. Then we consider the well-known characterization of the Veronese surface, due to Severi, as the unique non-degenerate surface in \(\mathbb{P}^{N}\), N ≥ 5, not a cone and with \(\dim (SX) = 4\). The novelty of our proof of this result and of its generalization to arbitrary dimension in Theorem 3.4.4 relies on a complete parallel with Mori’s proof of the characterization of the projective space as the unique manifold with ample tangent bundle.

The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine-linear. In this paper, we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three-dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that [Formula: see text] fixed projective points in real [Formula: see text]-dimensional projective space, through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.

Grassmann codes are linear codes associated with the Grassmann variety $G(\ell,m)$ of $\ell$-dimensional subspaces of an $m$ dimensional vector space $\mathbb{F}_{q}^{m}.$ They were studied by Nogin for general $q.$ These codes are conveniently described using the correspondence between non-degenerate $[n,k]_{q}$ linear codes on one hand and non-degenerate $[n,k]$ projective systems on the other hand. A non-degenerate $[n,k]$ projective system is simply a collection of $n$ points in projective space $\mathbb{P}^{k-1}$ satisfying the condition that no hyperplane of $\mathbb{P}^{k-1}$ contains all the $n$ points under consideration. In this paper we will determine the weight of linear codes $C(3,8)$ associated with Grassmann varieties $G(3,8)$ over an arbitrary finite field $\mathbb{F}_{q}$. We use a formula for the weight of a codeword of $C(3,8)$, in terms of the cardinalities certain varieties associated with alternating trilinear forms on $\mathbb{F}_{q}^{8}.$ For $m=6$ and $7,$ the weight spectrum of $C(3,m)$ associated with $G(3,m),$ have been fully determined by Kaipa K.V, Pillai H.K and Nogin Y. A classification of trivectors depends essentially on the dimension $n$ of the base space. For $n\leq 8 $ there exist only finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\mathbb{\bar{F}}$ to $\mathbb{F}.$ This program is partially determined by Noui L. and Midoune N. and the classification of trilinear alternating forms on a vector space of dimension $8$ over a finite field $\mathbb{F}_{q}$ of characteristic other than $2$ and $3$ was solved by Noui L. and Midoune N. We describe the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$, where char $\mathbb{F}_{q}\neq 3.$ This fact we use to determine the weight of the $\mathbb{F}_{q}$-forms.

Let μ and ν be two gaussian measures on a vector space E, μ and ν their canonical images on the projective space ℙ(E). We show that either μ and ν are mutually singular or there exists a homothety h in E such that h(μ) is equivalent to ν. We gather some properties of a mean M(a 1,..., a n ) natural in this context.

Cycle indices of linear, affine, and projective groups

Let \({L = \mathbb{F}_{q^n}}\) be a finite field and let \({F=\mathbb{F}_q}\) be a subfield of L. Consider L as a vector space over F and the associated projective space that is isomorphic to PG(n − 1, q). The properties of the projective mapping induced by \({x \mapsto x^{-1}}\) have been studied in Csajbok (Finite Fields Appl. 19:55–66, 2013), Faina et al. (Eur. J. Comb. 23:31–35, 2002), Havlicek (Abh. Math. Sem. Univ. Hamburg 53:266–275, 1983), Herzer (Abh. Math. Sem. Univ. Hamburg 55:211–228 1985, Handbook of Incidence Geometry. Buildings and Foundations. Elsevier, Amsterdam, 1995). The image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer k, if q ≥ 2k − 1, then there are partitions of PG(2k − 1, q) in normal rational curves of degree 2k − 1. For smaller q the same construction gives partitions in (q + 1)-tuples of independent points.