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.

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 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.

Maximal Strictly Partial Spreads

1978 ◽  
Vol 30 (03) ◽  
pp. 483-489 ◽  
Author(s):  
Gary L. Ebert
Keyword(s):  
Projective Space ◽  
Partial Spread ◽  
3 Dimensional ◽  
Partial Spreads ◽  
Maximal Partial Spread ◽  
Dimensional Projective Space

Let ∑ = PG(3, q) denote 3-dimensional projective space over GF(q). A partial spread of ∑ is a collection W of pairwise skew lines in ∑. W is said to be maximal if it is not properly contained in any other partial spread. If every point of ∑ is contained in some line of W, then W is called a spread. Since every spread of PG(3, q) consists of q2 + 1 lines, the deficiency of a partial spread W is defined to be the number d = q2 + 1 — |W|. A maximal partial spread of ∑ which is not a spread is called a maximal strictly partial spread (msp spread) of ∑.

CHARACTERIZING HERMITIAN VARIETIES IN THREE- AND FOUR-DIMENSIONAL PROJECTIVE SPACES

2018 ◽  
Vol 107 (1) ◽  
pp. 1-8 ◽  
Author(s):  
ANGELA AGUGLIA
Keyword(s):  
Projective Spaces ◽  
Intersection Numbers

We characterize Hermitian cones among the surfaces of degree$q+1$of$\text{PG}(3,q^{2})$by their intersection numbers with planes. We then use this result and provide a characterization of nonsingular Hermitian varieties of$\text{PG}(4,q^{2})$among quasi-Hermitian ones.

scholarly journals Proving and Generalizing Desargues’ Two-Triangle Theorem in 3-Dimensional Projective Space

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Dimitrios Kodokostas
Keyword(s):  
Projective Space ◽  
Arbitrary Number ◽  
Three Dimensional ◽  
3 Dimensional ◽  
Dimensional Projective Space ◽  
Set Of Points

With the use of only the incidence axioms we prove and generalize Desargues’ two-triangle Theorem in three-dimensional projective space considering an arbitrary number of points on each one of the two distinct planes allowing corresponding points on the two planes to coincide and three points on any of the planes to be collinear. We provide three generalizations and we define the notions of a generalized line and a triangle-connected plane set of points.

scholarly journals LINKING NUMBER IN A PROJECTIVE SPACE AS THE DEGREE OF A MAP

2007 ◽  
Vol 16 (04) ◽  
pp. 489-497 ◽  
Author(s):  
JULIA VIRO DROBOTUKHINA
Keyword(s):  
Projective Space ◽  
Configuration Space ◽  
The Self ◽  
Real Projective Space ◽  
3 Dimensional ◽  
Linking Number ◽  
Degree Of A Map ◽  
Number Of Cycles ◽  
Dimensional Projective Space

For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of cycles in a projective space of arbitrary odd dimension and the self-linking number of a zero homologous knot in the 3-dimensional projective space.

Constraint and Singularity Analysis of Lower-Mobility Parallel Manipulators With Parallelogram Joints

2010 ◽  
Author(s):  
Semaan Amine ◽  
Daniel Kanaan ◽  
Ste´phane Caro ◽  
Philippe Wenger
Keyword(s):  
Projective Space ◽  
Screw Theory ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
3 Dimensional ◽  
Geometric Conditions ◽  
Constraint Analysis ◽  
Lower Mobility ◽  
Dimensional Projective Space ◽  
Grassmann Cayley Algebra

This paper presents a general approach to analyze the singularities of lower-mobility parallel manipulators with parallelogram joints. Using screw theory, the concept of twist graph is introduced and the twist graphs of two types of parallelogram joints are established in order to simplify the constraint analysis of the manipulators under study. Using Grassmann-Cayley Algebra, the geometric conditions associated with the dependency of six Plu¨cker vectors of finite and infinite lines in the 3-dimensional projective space are reformulated in the superbracket in order to derive the geometric conditions for parallel singularities. The methodology is applied to three lower-mobility parallel manipulators with parallelogram joints: the Delta-linear robot, the Orthoglide robot and the H4 robot. The geometric interpretations of the singularities of these robots are given.

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