scholarly journals Canonical and log canonical thresholds of multiple projective spaces

2019 ◽  
Author(s):  
Aleksandr V. Pukhlikov
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Regularity Conditions ◽  
Large Classes ◽  
Log Canonical Threshold ◽  
Birational Rigidity ◽  
Log Canonical ◽  
Finite Set ◽  
Dimensional Projective Space ◽  
Almost All

AbstractWe show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where $$d\geqslant 4$$d⩾4, is equal to 1 for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano–Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

scholarly journals Cofinitely and co-countably projective spaces

2002 ◽  
Vol 3 (2) ◽  
pp. 185
Author(s):  
Pablo Mendoza Iturralde ◽  
Vladimir V. Tkachuk
Keyword(s):  
Topological Group ◽  
Projective Space ◽  
Infinite Family ◽  
Projective Spaces ◽  
Finite Union ◽  
Paracompact Space ◽  
Finite Set ◽  
Discrete Spaces

<p>We show that X is cofinitely projective if and only if it is a finite union of Alexandroff compactatifications of discrete spaces. We also prove that X is co-countably projective if and only if X admits no disjoint infinite family of uncountable cozero sets. It is shown that a paracompact space X is co-countably projective if and only if there exists a finite set B C X such that B C U ϵ τ (X) implies │X\U│ ≤ ω. In case of existence of such a B we will say that X is concentrated around B. We prove that there exists a space Y which is co-countably projective while there is no finite set B C Y around which Y is concentrated. We show that any metrizable co-countably projective space is countable. An important corollary is that every co-countably projective topological group is countable.</p>

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

2012 ◽  
Vol 26 (27n28) ◽  
pp. 1243013
Author(s):  
METOD SANIGA
Keyword(s):  
Black Hole ◽  
Projective Space ◽  
Projective Spaces ◽  
Quadric Surface ◽  
Hermitian Variety ◽  
Finite Projective Spaces ◽  
New Perspective ◽  
Dimensional Projective Space ◽  
Nondegenerate Quadric

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.

scholarly journals A Characterization of the Hermitian Variety in Finite 3-Dimensional Projective Spaces

10.37236/3416 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Vito Napolitano
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Intersection Numbers ◽  
Combinatorial Characterization ◽  
Hermitian Variety ◽  
3 Dimensional ◽  
Dimensional Projective Space

A combinatorial characterization of a non-singular Hermitian variety of the finite 3-dimensional projective space via its intersection numbers with respect to lines and planes is given.   A corrigendum was added on March 29, 2019.

Branched Coverings and Minimal Free Resolution for Infinite-Dimensional Complex Spaces

2003 ◽  
Vol 10 (1) ◽  
pp. 37-43
Author(s):  
E. Ballico
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Free Resolution ◽  
Complete Intersections ◽  
Vanishing Theorem ◽  
Sheaf Cohomology ◽  
Minimal Free Resolution ◽  
Branched Coverings ◽  
Infinite Dimensional ◽  
Complex Spaces

Abstract We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1. In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P(V ).

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

Transformations depending on sets of associated points

1952 ◽  
Vol 48 (3) ◽  
pp. 383-391
Author(s):  
T. G. Room
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Three Dimensional ◽  
Cubic Curves ◽  
Birational Transformations ◽  
Quadric Surfaces ◽  
Base Points ◽  
Dimensional Projective Space ◽  
Quartic Curves

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

The geometry and algebra of the representations of the Lorentz group

1968 ◽  
Vol 303 (1474) ◽  
pp. 381-396 ◽  
Keyword(s):  
Projective Space ◽  
Lorentz Group ◽  
Point Of View ◽  
Normal Curve ◽  
Geometrical Representation ◽  
Spinor Space ◽  
Dimensional Projective Space ◽  
And Algebra ◽  
Rational Normal Curve ◽  
Formal Point

In this paper a (2j + l)-spinor analysis is developed along the lines of the 2-spinor and 3-spinor ones. We define generalized connecting quantities A μv (j) which transform like (j, 0) ⊗ (j -1, 0) in spinor space and like second rank tensors under transformations in space-time. The general properties of the A uv are investigated together with algebraic relations involving the Lorentz group generators, J μv . The connexion with 3j symbols is discussed. From a purely formal point of view we introduce a geometrical representation of a (2j +1)-spinor as a point in a 2j dimensional projective space. Then, for example, the charge con­jugate of a (2j + l)-spinor is just the polar of the corresponding point with respect to a certain rational, normal curve in the projective space. It is suggested that this representation will prove useful.

scholarly journals Projective Reconstruction in Algebraic Vision

2019 ◽  
Vol 63 (3) ◽  
pp. 592-609
Author(s):  
Atsushi Ito ◽  
Makoto Miura ◽  
Kazushi Ueda
Keyword(s):  
Projective Space ◽  
Arbitrary Dimension ◽  
Projective Spaces ◽  
Rational Maps ◽  
Projective Reconstruction ◽  
Reconstruction Theorem

AbstractWe discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.

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