projective geometry
1 Subspace geometries
Given a vector space^{} $V$, $dimV>0$, the projective geometry^{} of $V$ is the set of all subspaces^{} of $V$ ordered by set inclusion. It is typically denoted $PG(V)$ or $P(V)$. The vector space may be over a field or a division ring.
The partially ordered set^{} (poset) of all subspaces of a vector space is a geometric lattice. So next to a boolean lattice, it is one of the best lattices one could expect. However, an alternative to viewing projective geometry $PG(V)$ as a lattice^{} is one of viewing $PG(V)$ as a geometry^{}. For this we assign the to $PG(V)$.
 Points

Points in projective geometry are the 1dimensional subspaces. These are often denoted $PG{(V)}_{0}$ and are the atoms of the lattice. $PG{(V)}_{0}$ is often referred to as the projective space^{} of $V$ and denoted $k{P}^{n}$ where $V={k}^{n+1}$, especially in topological settings. Common examples include $\mathbb{R}{P}^{n}$, $\u2102{P}^{n}$. (More on projective spaces (http://planetmath.org/ProjectiveSpace).)
 Lines

As in the usual Euclidean geometry two distinct points should determine a unique line in the geometry. As two linearly independent^{} vectors span a plane in $V$, in to make two points in the geometry $PG(V)$ determine a line, we must define a line in projective space to a 2dimensional subspace.
Remark 1.
There is a conflict of terminology at this stage. Often it is useful to define a projective space as the set of all lines through the origin in a vector space $V$. But if we elect to view a projective space as a projective geometry then the lines in this definition correspond to points in the geometry, and lines are now the planes of the vector space containing the origin. Sometimes for clarity the phrase “projective point” and “projective line” can be used to resolve ambiguity.
 Planes

A projective plane^{} for a vector space $V$ is a 3dimensional subspace. The study of projective planes extends beyond the consideration of the poset of vector spaces however, and is the beginning of many interesting combinatorial problems. See the following section^{} on nonDesarguesian planes for further details.
 Hyperplanes^{}

Hyperplanes are subspaces of $V$, sometimes called codimension 1 subspaces. If $V$ is finite dimensional then points and hyperplanes are in a 11 correspondence. This correspondence leads to many situations where an exchange of a point with a hyperplane is considered. The simplest of these exchanges occurs through the notion of a perpendicular^{} subspace, i.e.: one may say a point is perpendicular to an entire hyperplane, and a hyperplane is perpendicular to just one point.
When $dimV=2$ then every point is a hyperplane which leads to many degenerate properties causing $dimV\ne 2$ conditions in many theorems of projective geometry.
2 Dual Geometries
Given a finite dimensional vector space $V$, the dual space^{} ${V}^{*}$ of all linear functionals^{} is isomorphic^{} as a vector space but it is also possible to $PG(V)$ to $PG({V}^{*})$ in a dual manner.
For every subspace $W$ of $V$ define
$${W}^{\bullet}=\{f\in {V}^{*}:(W)f=0\}.$$ 
That is, ${W}^{\bullet}$ is the set of all functionals^{} which contain $W$ in their kernel (nullspace^{}). It follows $dim{W}^{\bullet}=dimVdimW$ and the map from $PG(V)$ to $PG({V}^{*})$ determined by
$$W\mapsto {W}^{\bullet}$$ 
is orderreversing. It is also evident that under the natural isomorphism of $V\cong {V}^{**}$ we can further take $W={W}^{\bullet \bullet}$.
3 Morphisms of Projective Geometry
Morphisms from one projective geometry to another are defined as orderpreserving maps, also called projectivities^{}. In some context orderreversing maps may also be included which leads to the study of dualities and polarities^{}.
Given an orderreversing map $PG(V)\to PG(W)$, the map $PG(V)\to PG({W}^{*})$ determines a canonical orderpreserving map so that one can indeed consider simply the orderpreserving maps between projective geometries.
A collineation^{} is a function which maps any three collinear points (i.e.: three 1 dimensional subspaces which all lie in a single 2dimensional subspace) to three collinear points. These determine a unique orderpreserving map between the two projective geometries. Thus the morphisms of projective geometry are often identified with collineations. This is preferable when authoring theorems in the of geometry.
Remark 2.
Some authors prefer collineations to any projectivity $P\mathit{}G\mathit{}\mathrm{(}V\mathrm{)}\mathrm{\to}P\mathit{}G\mathit{}\mathrm{(}V\mathrm{)}$. Although there is no uniformity in these definitions, each takes collinear^{} triples to collinear triples thus preserving the geometries under consideration.
4 Notations for projective geometries
When the dimension $d$ of $V$ is finite we may write $PG(d1,k)$ where $k$ is the field (or division ring) of the vector space. Notice that the $d1$ indicates the dimension of the geometry, not the dimension of the vector space, though one can be attained from the other.
When $k$ is real, or complex, $PG(1,k)$ is often denoted $\mathbb{R}{P}^{1}$ and $\u2102{P}^{1}$ instead. Once again the $1$ denotes the dimensions of the geometry, and in this case also the manifold, not the vector space from which it is derived.
When $k=q$ we may further write $PG(d,q)$.
5 $PG(1,k)$
The 1 dimensional vector spaces have points and hyperplanes, but every point is a hyperplane. Every permutation^{} of points is a collineation. So the projective line is exceptional in many ways. Including in the fundamental theorem of projective geometry^{}.
5.1 Abstract Projective Geometry
Projective geometry in general can be axiomatized as achieved by Hilbert. The axioms precisely characterize the subspace lattice of a finite dimensional vector space, but the converse^{} is not generally true. Indeed, already for 1dimensional geometries, so called projective lines, i.e.: a set of points, it is clear that not all such geometries can be captured as the subspaces of a vector space. For example, there is no vector space with exactly 2 one dimensional subspaces. Such geometries however are of little interest as a geometry to themselves (though they are pivotal as subgeometries) for they have no .
When an abstract geometry is infinite dimensional or 2dimensional, it is possible that it is not isomorphic to the geometry of subspaces of any vector space. These geometries are termed “nonDesarguesian” as they do not carry with them a version of Desargues’ theorem. These geometries are a rich area of study, especially so called projective planes – geometries of dimension 2. Despite not having the structure^{} of a subspace geometry, so far every nonDesarguesian projective plane has still had order ${p}^{k}$ for some prime $p$. This has lead to the following unsolved problem:
Are their any projective planes of order not a power of a prime?
References
 1 Gruenberg, K. W. and Weir, A.J. Linear Geometry 2nd Ed. (English) [B] Graduate Texts in Mathematics. 49. New York  Heidelberg  Berlin: SpringerVerlag. (1977), pp. x198.
 2 Kantor, W. M. Lectures notes on Classical Groups^{}.
 3 Taylor, Donald E. The geometry of the classical groups Sigma Series in Pure Mathematics. 9. Heldermann Verlag, Berlin, xii+229, (1992), ISBN 3885380099.
Title  projective geometry 
Canonical name  ProjectiveGeometry 
Date of creation  20130322 15:51:25 
Last modified on  20130322 15:51:25 
Owner  Algeboy (12884) 
Last modified by  Algeboy (12884) 
Numerical id  26 
Author  Algeboy (12884) 
Entry type  Definition 
Classification  msc 51A10 
Classification  msc 51A30 
Classification  msc 51A05 
Classification  msc 51D25 
Related topic  PolaritiesAndForms 
Related topic  SesquilinearFormsOverGeneralFields 
Related topic  Perspectivity^{} 
Defines  projective geometry 