Linear sets in finite projective spaces

In this paper linear sets of finite projective spaces are studied and the “dual” of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides “old” results, new ones are proven and some open questions are discussed.     

Linear spaces with many projective planes

We assume that in a linear space there is a non-empty set M of points with the property that every plane containing a point of M is a projective plane. In section 3 an example is given that in general is not a projective space. But if M can be completed by two points to a generating set of P, then is a projective space.     

Linear programming bounds for codes in infinite projective spaces

We develop a technique for improving the universal linear programming bounds on the cardinality and the minimum distance of codes in projective spaces . We firstly investigate test functions P_j(m,n,s) having the property that P_j(m,n,s)<0 for somej if and only if the corresponding universal linear programming bound can be further improved by linear programming. Then we describe a method for improving the universal bounds. We also investigate the possibilities for attaining the first universal bounds.     

Limits of directed projective systems of probability spaces

Summary The problem of limits of directed projective systems of probability spaces is treated from the categorical point of view, with certain equivalence classes of measurable measurepreserving mappings, or of regular conditional probabilities playing the role of morphisms. A. o. the existence of limits of such systems is established under the condition that the spaces carry compact generating pavings, without invoking conditions like Bochners' sequential maximality. There are some side results on liftings and general martingales.This research was supported in part by a grant from the National Research Council of Canada.     

Monads on projective spaces

Let be a vector bundle on P ⁿ . There is a strong relationship between and its intermediate cohomology modules. In the case where has low rank, we exploit this relationship to provide various splitting criteria for . In particular, we give a splitting criterion for in terms of the vanishing of certain intermediate cohomology modules. We also show that the Horrocks-Mumford bundle is the only non-split rank two bundle on P ⁴ with a Buchsbaum second cohomology module.Partially supported by NSF Grants.     

Geodesics on weighted projective spaces

We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can “hear” the weights of a weighted projective space.     

Graphical operations on projective spaces

Motivated by the work of Crapo and Rota [6] on the lifting of a projective complex, we introduce a class of invariant operations associated to integral-weighted graphs, which we call graphical operations. Such operations generalize the sixth harmonic of a quadranguler set on a projective line. We determine the expansion of the graphical operations in terms of multi-linear bracket polynomials in a Grassmann-Cayley algebra. Reducibility and compositions of such invariant operations are also investigated with a number of examples.Supported by Courant Instructorship, New York University.     

On canonical embeddings of complex projective spaces in real projective spaces

In this paper, we construct a natural embedding \(\sigma :\mathbb{C}P_\mathbb{R}^{n} \to \mathbb{R}P^{n^2 + 2n} \) of the complex projective space ?P ⁿ considered as a 2n-dimensional, real-analytic manifold in the real projective space \(\mathbb{R}P^{n^2 + 2n} \). The image of the embedding σ is called the ?P ⁿ-surface. To construct the embedding, we consider two equivalent approaches. The first approach is based on properties of holomorphic bivectors in the realification of a complex vector space. This approach allows one to prove that a ?P-surface is a flat section of a Grassman manifold. In the second approach, we use the adjoint representation of the Lie group U(n + 1) and the canonical decomposition of the Lie algebra u(n). This approach allows one to state a gemetric characterization of the canonical decomposition of the Lie algebra u(n). Moreover, we study properties of the embedding constructed. We prove that this embedding determines the canonical Kähler structure on ?P _? ⁿ . In particular, the Fubini-Study metric is exactly the first fundamental form of the embedding and the complex structure on ?P _? ⁿ is completely defined by its second fundamental form; therefore, this embedding is said to be canonical. Moreover, we describe invariant and anti-invariant completely geodesic submanifolds of the complex projective space.     

Linear chaotic systems in generalized Fock-Bargmann spaces

In this paper we consider dynamical systems generated by iterations of linear operators which act as annihilation operators in generalized Fock-Bargmann spaces. It is proved that these systems are chaotic in the sense of Devaney.     

Parallelisms of projective spaces

New and old results on parallelisms of projective spaces are surveyed.     

Beilinson resolutions on weighted projective spaces

Beilinson's theorem [Funct. Anal. Appl. 12 (1978) 214–216], which describes the bounded derived category of coherent sheaves on $\text{P}{}^{\mathrm{n}}$, is extended to weighted projective spaces. This result is obtained by considering, instead of the usual category of coherent sheaves, a suitable category of graded coherent sheaves (which is equivalent in the case of $\text{P}{}^{\mathrm{n}}$). To cite this article: A. Canonaco, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Maps into projective spaces

We compute the cohomology of the Picard bundle on the desingularization $\tilde{J}^d(Y)$ of the compactified Jacobian of an irreducible nodal curve Y. We use it to compute the cohomology classes of the Brill–Noether loci in $\tilde{J}^d(Y)$ . We show that the moduli space M of morphisms of a fixed degree from Y to a projective space has a smooth compactification. As another application of the cohomology of the Picard bundle, we compute a top intersection number for the moduli space M confirming the Vafa–Intriligator formulae in the nodal case.     

On projective symmetries on Finsler spaces

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of I-invariant projective vector fields. The sub-algebra of the C-projective vector fields, leaving the H-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature.     

Categories of projective spaces

Starting with an abelian category , a natural construction produces a category such that, when is an abelian category of vector spaces, is the corresponding category of projective spaces. The process of forming the category destroys abelianess, but not completely, and the precise measure of what remains of it gives the possibility to reconstruct out from , and allows to characterize categories of the form , for an abelian (projective categories). The characterization is given in terms of the notion of “Puppe exact category” and of an appropriate notion of “weak biproducts”. The proof of the characterization theorem relies on the theory of “additive relations”.