Dimensional Projective Space Research Articles

Segre embeddings and finite semifields

Each embedded product space PG(n,q)×PG(n,q) in an (n2+n−1)-dimensional projective space is obtained by projecting the Segre variety Sn,n,q from an n-subspace δ skew with its first secant variety (Zanella, 1996 [22]). On the other hand, when δ is skew with the (n−1)-th secant variety, it determines a semifield of order qn+1 whose center contains Fq (Lavrauw, 2011 [17]). A relationship arises between a particular class of embeddings of PG(n,q)×PG(n,q) in PG(n2+n−1,q) and semifields of the above type. For this reason, such embeddings will be called semifield embeddings. In this paper we establish this connection and prove that projectively equivalent semifield embeddings that do not exchange subspaces of different kind are related to isotopic semifields, and conversely. Exchanging the order in the product leads to the transition from a semifield to its transpose.

L–S categories of vector bundles over projective spaces

I.M. James calculated vecat(kγR), the L–S category of kγR, for k=2t and γR, the canonical line bundle over the (2t−1)-dimensional real projective space. In this paper, we calculate vecat(kγR) for integers k and γR, the canonical line bundle over projective spaces. We also calculate vecatFξ, the L–S category of ξ as an F-vector bundle, where F=R or C, for ξ, multiple Whitney sums of canonical line bundles over complex and quaternion projective spaces.

Glicci ideals

AbstractA central problem in liaison theory is to decide whether every arithmetically Cohen–Macaulay subscheme of projective $n$-space can be linked by a finite number of arithmetically Gorenstein schemes to a complete intersection. We show that this can indeed be achieved if the given scheme is also generically Gorenstein and we allow the links to take place in an $(n+ 1)$-dimensional projective space. For example, this result applies to all reduced arithmetically Cohen–Macaulay subschemes. We also show that every union of fat points in projective 3-space can be linked in the same space to a union of simple points in finitely many steps, and hence to a complete intersection in projective 4-space.

A combinatorial characterization of the Hermitian surface

In this paper we prove that in a projective space of dimension three and square order q2 a (q5+q3+q2+1)-set of class [1,2,…,q+1,q2+1]1 and type (m,n)2 is a Hermitian surface.

Periodic Structure of Transversal Maps on $$\mathbb{C }$$ P $$^{n}$$ , $$\mathbb{H }$$ P $$^{n}$$ and $$\mathbb{S }^{p}\times \mathbb{S }^{q}$$

A \(\mathcal{C }^{1}\) map \(f:\mathbb{M }\rightarrow \mathbb{M }\) is called transversal if for all \(m\in \mathbb{N }\) the graph of \(f^{m}\) intersects transversally the diagonal of \(M\times M\) at each point \((x,x)\) being \(x\) a fixed point of \(f^m\). Let \(\mathbb{C }\)P\(^{n}\) be the \(n\)-dimensional complex projective space, \(\mathbb{H }\)P\(^{n}\) be the \(n\)-dimensional quaternion projective space and \(\mathbb{S }^{p}\times \mathbb{S }^{q}\) be the product space of the \(p\)-dimensional with the \(q\)-dimensional spheres, \(p\ne q\). Then for the cases \(\mathbb{M }\) equal to \(\mathbb{C }\)P\(^{n}\), \(\mathbb{H }\)P\(^{n}\) and \(\mathbb{S }^{p}\times \mathbb{S }^{q}\) we study the set of periods of \(f\) by using the Lefschetz numbers for periodic points.

Isospin particle systems on quaternionic projective spaces

We construct the isospin particle system on $n$-dimensional quaternionic projective spaces in the presence of the Belavin-Polyakov-Schwarz-Tyupkin instanton by the reduction from the free particle on $(2n+1)$-dimensional complex projective space. Then we add to this system a ``quaternionic oscillator potential'' and show that this oscillatorlike system is superintegrable. We show that, besides the analogs of quadratic constants of motion of the spherical (Higgs) and $\mathbb{C}{P}^{n}$ oscillators, it possesses the third-order constants of motion, which are functionally independent from the quadratic ones.

The variety of polar simplices

A collection of n distinct hyperplanes L_i = {l_i=0} \subset {\mathbf P}^{n-1} , the (n-1) -dimensional projective space over an algebraically closed field of characteristic not equal to 2, is a polar simplex of a smooth quadric Q^{n-2}={q=0} , if each L_i is the polar hyperplane of the point p_i = \bigcap_{j \ne i} L_j , equivalently, if q= l_1^2+\ldots+l_n^2 for suitable choices of the linear forms l_i . In this paper we study the closure VPS(Q,n) \subset \mathrm{Hilb}_{n}(\check{\mathbf P}^{n-1}) of the variety of sums of powers presenting Q from a global viewpoint: VPS(Q,n) is a smooth Fano variety of index 2 and Picard number 1 when n<6 , and VPS(Q,n) is singular when n\geq 6 .

Simplicial resolutions and spaces of algebraic maps between real projective spaces

We show that the space A˜d(m,n) consisting of all real projective classes of (n+1)-tuples of real coefficients homogeneous polynomials of degree d in (m+1) variables, without common real roots except zero, has the same homology as the space Map(RPm,RPn) of continuous maps from the m-dimensional real projective space RPm into the n real dimensional projective space RPn up to dimension (n−m)(d+1)−1. This considerably improves the main result of Adamaszek et al. (2011) [1].

FINITE PROJECTIVE SPACES, GEOMETRIC SPREADS OF LINES AND MULTI-QUBITS

Given a (2N-1)-dimensional projective space over GF(2), PG (2N-1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG (N-1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG (N-1, 4). Under such mapping, a nondegenerate quadric surface of the former space has for its image a nonsingular Hermitian variety in the latter space, this quadric being hyperbolic or elliptic in dependence on N being even or odd, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric Pauli operators into a new perspective. The N = 4 case is taken to illustrate the issue, due to its link with the so-called black-hole/qubit correspondence.

Estimation of the minimal number of periodic points for smooth self-maps of odd dimensional real projective spaces

Let f be a smooth self-map of a closed connected manifold of dimension m⩾3. The authors introduced in [G. Graff, J. Jezierski, Minimizing the number of periodic points for smooth maps. Non-simply connected case, Topology Appl. 158 (3) (2011) 276–290] the topological invariant NJDr[f], where r is a fixed natural number, which is equal to the minimal number of r-periodic points in the smooth homotopy class of f. In this paper smooth self-maps of real projective space RPm, where m>3 is odd, are considered and the estimations from below and above for NJDr[f] are given.

A characterization of the non-Baer lines of a Hermitian surface

We give a characterization of the set of the lines which either belong to or are tangent to a non-singular Hermitian surface in the projective space of dimension 3 and order q 2.

A realization theorem for modules of constant Jordan type and vector bundles

Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. We introduce functors F_i from finitely generated kE-modules of constant Jordan type to vector bundles over projective space of dimension r-1. The fibers of these functors encode complete information about the Jordan type of the module. We prove that given any vector bundle of rank s on P^{r-1}, there is a kE-module M of stable constant Jordan type [1]^s such that the functor F_1 applied to M yields the original vector bundle for p=2 and the Frobenius twist of the original vector bundle for p>2. We also prove that the theorem cannot be improved if p is odd, because if M is any module of stable constant Jordan type [1]^s then the Chern numbers c_1, ... ,c_{p-2} of F_1(M) are divisible by p.

Subclass discriminant Nonnegative Matrix Factorization for facial image analysis

Nonnegative Matrix Factorization (NMF) is among the most popular subspace methods, widely used in a variety of image processing problems. Recently, a discriminant NMF method that incorporates Linear Discriminant Analysis inspired criteria has been proposed, which achieves an efficient decomposition of the provided data to its discriminant parts, thus enhancing classification performance. However, this approach possesses certain limitations, since it assumes that the underlying data distribution is unimodal, which is often unrealistic. To remedy this limitation, we regard that data inside each class have a multimodal distribution, thus forming clusters and use criteria inspired by Clustering based Discriminant Analysis. The proposed method incorporates appropriate discriminant constraints in the NMF decomposition cost function in order to address the problem of finding discriminant projections that enhance class separability in the reduced dimensional projection space, while taking into account subclass information. The developed algorithm has been applied for both facial expression and face recognition on three popular databases. Experimental results verified that it successfully identified discriminant facial parts, thus enhancing recognition performance.

On singular quadratic complexes, quintic curves and Cremona transformations

Let X be a quadratic complex given by the intersection of two nonsingular quadrics in a projective space of dimension five. Let L be a line contained in X, and π the projection from X to a projective three space with center L. When X is nonsingular the map π is birational and the base locus scheme of π−1 is a smooth quintic curve of genus 2. Now assume X is a singular irreducible and reduced quadratic complex and consider the same set up. The purpose of this work is to classify quintic curves arising as the base locus scheme of π−1 in the case where π is birational and the Cremona transformations obtained by composing π−1 with another projection of the same type.

Courbes algébriques ordinaires et tissus associés

Let Γ be a complex algebraic curve of degree d, non-degenerate, reduced, and irreducible, in the complex projective space Pn(n⩾3,d⩾n). Denoting by c(n,h) the dimension of the vector space of homogeneous polynomials of degree h with respect to n variables, let k0 be the integer (⩾1) such that c(n,k0)⩽d<c(n,k0+1). The curve Γ is said to be ordinary if it has the following property: the set of algebraic hypersurfaces of a “generic” hyperplane H of Pn which have degree h and contain the hyperplane section Γ∩H, is empty if h⩽k0, and is a projective space of dimension c(n,h)−d−1 if h>k0. Equivalently when k0⩾2, the associated web in Pˇn of such a curve is “ordinary” in the sense of Cavalier and Lehmann (2012) [1]. The arithmetic genus of an ordinary curve is upper-bounded by the number π′(n,d)=∑h=1k0(d−c(n,h))(=k0d−c(n+1,k0)+1), and this bound is reached for these ordinary curves which are arithmetically Cohen–Macaulay. For n=3 and any d⩾3, the family of the ordinary curves of degree d which are arithmetically Cohen–Macaulay is non-empty, and is an irreducible component of the Hilbert scheme Hd,π′(3,d).By contrast, the complete intersection of n−1 algebraic hypersurfaces is never ordinary if k0⩾2.

An Invariant for Hypersurfaces in Prime Characteristic

A hypersurface of order $(n+1)(p^h-1)$ in projective space of dimension $n$ of prime characteristic $p$ has an invariant monomial. This implies that a hypersurface of order $(n+1)(p^h-1)-1$ determines an invariant point. A hypersurface of order $d<n+1$ in a projective space of dimension $n$ of characteristic two has an invariant set of subspaces of dimension $d-1$ determined by one linear condition on the Grassmann coordinates of the dual subspaces.

Kähler immersions of homogeneous Kähler manifolds into complex space forms

In this paper we study the homogeneous Kaehler manifolds (h.K.m.) which can be Kaehler immersed into finite or infinite dimensional complex space forms. On the one hand we completely classify the h.K.m. which can be Kaehler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. On the other hand, we extend known results about Kaehler immersions into the finite dimensional complex projective space to the infinite dimensional setting

Geometry of nonholonomic complexes NGr (1, 5, 5)

In this article the differential geometry of nonholonomic complexes in five dimensional projective space is considered. The algebraic structure of the first fundamental object is considered and the geometrcal interpretations of some comitants of the first fundamental object are given.&#x0D; &#x0D;

Maps into projective spaces: Liquid crystal and conformal energies

Variational problems for the liquid crystal energy of mappings from three-dimensional domains into the real projective plane are studied. More generally, we study the dipole problem, the relaxed energy, and density properties concerning the conformal $p$-energy of mappings from $n$-dimensional domains that are constrained to take values into the $p$-dimensional real projective space, for any positive integer $p$. Furthermore, a notion of optimally connecting measure for the singular set of such class of maps is given.

Generalization of a theorem of Clunie and Hayman

Clunie and Hayman proved that if the spherical derivative ‖f ′‖ of an entire function satisfies ‖f ′‖(z) = O(|z|σ) then T (r, f) = O(rσ+1). We generalize this to holomorphic curves in projective space of dimension n omitting n hyperplanes in general position. MSC 32Q99, 30D15. Introduction We consider holomorphic curves f : C → P; for the general background on the subject we refer to [7]. The Fubini–Study derivative ‖f ‖ measures the length distortion from the Euclidean metric in C to the Fubini–Study metric in P. The explicit expression is ‖f ‖ = ‖f‖ ∑ i<j |f ′ ifj − fif ′ j| , where (f0, . . . , fn) is a homogeneous representation of f (that is the fj are entire functions which never simultaneously vanish), and ‖f‖ = n