Blocking Sets and Skew Subspaces of Projective Space

1980 ◽  
Vol 32 (3) ◽  
pp. 628-630 ◽  
Author(s):  
Aiden A. Bruen
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Projective Spaces ◽  
Higher Dimensions ◽  
Blocking Set ◽  
Blocking Sets ◽  
Partial Spreads ◽  
Dimension One

In what follows, a theorem on blocking sets is generalized to higher dimensions. The result is then used to study maximal partial spreads of odd-dimensional projective spaces.Notation. The number of elements in a set X is denoted by |X|. Those elements in a set A which are not in the set Bare denoted by A — B. In a projective space Σ = PG(n, q) of dimension n over the field GF(q) of order q, ┌d(Ωd, Λd, etc.) will mean a subspace of dimension d. A hyperplane of Σ is a subspace of dimension n — 1, that is, of co-dimension one.A blocking set in a projective plane π is a subset S of the points of π such that each line of π contains at least one point in S and at least one point not in S. The following result is shown in [1], [2].

Blocking Sets in Projective Spaces

1978 ◽  
Vol 30 (4) ◽  
pp. 856-862
Author(s):  
Gary L. Ebert
Keyword(s):  
Game Theory ◽  
Projective Plane ◽  
Projective Spaces ◽  
Blocking Set ◽  
Blocking Sets ◽  
Partial Spreads ◽  
Image Position

Blocking sets in projective spaces have been of interest for quite some time, having applications to game theory (see [6; 7]) as well as finite nets and partial spreads (see [5]). In [4] Bruen showed that if B is a blocking set in a projective plane of order n, then

scholarly journals Homography in ℝℙ

2016 ◽  
Vol 24 (4) ◽  
pp. 239-251 ◽  
Author(s):  
Roland Coghetto
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Projective Spaces ◽  
Invertible Matrix ◽  
Real Projective Plane ◽  
The Real ◽  
Mizar Mathematical Library ◽  
Plücker Relation

Summary The real projective plane has been formalized in Isabelle/HOL by Timothy Makarios [13] and in Coq by Nicolas Magaud, Julien Narboux and Pascal Schreck [12]. Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by Wojciech Leonczuk [9], Krzysztof Prazmowski [10] and by Wojciech Skaba [18]. In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Plücker relation in rank 3 [2], [1], [7], [16], [17]. Then we show that the projective space induced (in the sense defined in [9]) by ℝ3 is a projective plane (in the sense defined in [10]). Finally, in the real projective plane, we define the homography induced by a 3-by-3 invertible matrix and we show that the images of 3 collinear points are themselves collinear.

Partial Spreads and Replaceable Nets

1971 ◽  
Vol 23 (3) ◽  
pp. 381-391 ◽  
Author(s):  
A. Bruen
Keyword(s):  
Projective Plane ◽  
Blocking Set ◽  
Partial Spreads

A blocking set S in a projective plane Π is a subset of the points of Π such that every line of Π contains at least one point of S and at least one point not in S. In previous papers [5; 6], we have shown that if Π is finite of order n, then n + √n + 1 ≦ |S| ≦ n2 – √n (see [6, Theorem 3.9]), where |S| stands for the number of points of S. This work is concerned with some applications of the above result to nets and partial spreads, and with some examples of partial spreads which give rise to unimbeddable nets of small deficiency.In the next section we re-prove a well known result of Bruck which states that if N is a replaceable net of order n and degree k then k ≧ √n + 1, and show how this bound can be improved if n = 7, 8, or 11.

scholarly journals The Possibility of Applying Rumen Research at the Projective Plane PG (2,17)

2019 ◽  
Vol 13 (8) ◽  
pp. 150
Author(s):  
Shaymaa Haleem Ibrahim ◽  
Nada Yassen Kasm
Keyword(s):  
Projective Plane ◽  
Theoretical Method ◽  
Galois Field ◽  
Blocking Set ◽  
Blocking Sets ◽  
Projection Plane ◽  
Set Of Points

One of the main objectives of this research is to use a new theoretical method to find arcs and Blocking sets. This method includes the deletion of a set of points from some lines under certain conditions explained in a paragraph 2.In this paper we were able to improve the minimum constraint of the (256,16) – arc in the projection plane PG(2,17).Thus , we obtained a new {50,2}-blocking set for size Less than 3q , and according to the theorem (1.3.1),we obtained the linear 257,3,24117    code, theorem( 2.1.1 ) giving some examples on arcs of the Galois field GF(q);q=17."

AN ALGORITHM FOR CONSTRUCTING BLOCKING SETS FOR PROJECTIVE PLANES

1992 ◽  
Vol 02 (04) ◽  
pp. 437-442
Author(s):  
RUTH SILVERMAN ◽  
ALAN H. STEIN
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Proper Subset ◽  
Blocking Set ◽  
Blocking Sets ◽  
Unique Point ◽  
Unique Line ◽  
The Family ◽  
Property B

A family of sets is said to have property B(s) if there is a set, referred to as a blocking set, whose intersection with each member of the family is a proper subset of that blocking set and contains fewer than s elements. A finite projective plane is a construction satisfying the two conditions that any two lines meet in a unique point and any two points are on a unique line. In this paper, the authors develop an algorithm of complexity O(n3) for constructing a blocking set for a projective plane of order n.

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 ).

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).

scholarly journals Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

10.37236/2582 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tamás Héger ◽  
Marcella Takáts
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
The Other ◽  
Blocking Set ◽  
Metric Dimension ◽  
Desarguesian Projective Plane ◽  
Resolving Set ◽  
Incidence Graph ◽  
Vertex Set

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

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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