Projective Geometry - Lecture Notes Chapter 1

1.3 The Structure of Projective Geometry

Def: A subset of the point set is linear if for every pair of distinct points P and Q in , each point of the line PQ is in . I.e., (PQ), where if g is a line (g) denotes the set of points incident with g.

Any linear set of a projective space P is a (possibly degenerate) projective space. The induced space P() is a subspace of P.

Examples: , singletons, the set of points on a line, the set of all points of the space are linear sets.

Since the intersection of an arbitrary number of linear sets is a linear set, we can define, for any subset of points,

span() = <> := { | , a linear set}.
i.e., <> is the smallest linear set containing .

Def: A set of points is called collinear if all points of are incident with a common line. The set is called noncollinear if there is no line incident with all the points of .
The span of a set of three noncollinear points is called a plane.

Theorem 1.3.1: (Greedy Algorithm) Let be a nonempty linear set of P, and let P be a point of P. Then

<,P> = {(PQ) | Q }.
Furthermore, each line of <,P> intersects .

Theorem 1.3.2: (Exchange Property) Let be a linear set of P, and let P be a point of P that does not lie in . Then,

if Q<,P>\ then P <,Q>, hence <,P> = <,Q>.
(Any 2 distinct subspaces containing that are spanned by and a point outside of intersect only in the points of .)

Def: A set of points of P is called independent if for any s/set ' and any point P in \' we have P <>. (When P <\{P}> we say that is dependent.

Examples: Any singleton, any pair of points, 3 noncollinear points, or any subset of an independent set are independent sets.

Def: An independent set of P that spans P is called a basis of P.

Theorem 1.3.3: A set of P is a basis of P iff is a minimal spanning set.

Def: A projective space P is called finitely generated if there exists a finite set of points which span P.

In the following we shall assume that P is finitely generated.

Theorem 1.3.4: Let be a finite spanning set of P. Then there exists a basis of P such that . In particular, P has a finite basis.

Lemma 1.3.5: Let be an independent set of points of P, 1 and 2 subsets of . If is finite, then

<12> = <1> <2>.

Lemma 1.3.6: (Exchange Lemma) Let be a finite basis of P, and let P be a point of P. Then there is a point Q in with the property that the set

is also a basis of P. (skip proof)

Theorem 1.3.7: (Steinitz exchange theorem for projective spaces) Let be a finite basis of P, and let r = ||. If is an independent set having s points then we have:

  1. sr
  2. There is a subset * of with |*| = r - s such that * is a basis of P.
(skip proof)

Corollary 1.3.8: (Basis Extension Theorem) Let P be a finitely generated projective space. Any two bases of P have the same number of elements. Moreover, any independent set (in particular any basis of a subspace) can be extended to a basis of P.

Def: If the number of elements in a basis of P (and hence all bases) is d + 1, then d is called the dimension of P and denoted by dim(P).

Lemma 1.3.9: Let be a subspace of the finitely generated projective space P.

  1. dim() dim(P),
  2. dim() = dim(P) iff = P.

Def: Let P be a projective space of dimension d. Subspaces of dimension 2 are called planes and subspaces of dimension d-1 are called hyperplanes.
We denote the set of all subspaces of P by (P). We call (P) together with the subset relation the projective geometry belonging to the projective space P.

Def: The empty set and the whole space are called trivial subspaces. The set of all nontrivial subspaces is denoted by *(P).

Note that (*(P), ) is a geometry of rank d.

Lemma 1.3.10: Let P be a d-dimensional projective space and let U be a t-dimensional subspace of P (-1td). Then there exist d - t hyperplanes of P such that U is the intersection of these hyperplanes.

Theorem 1.3.11: (Dimension Formula) Let U and W be subspaces of P. Then

dim(<U,W>) = dim(U) + dim(W) - dim(UW).

Corollary 1.3.12: Let P be a projective space and let H be a hyperplane of P. Then for any subspace U of P either

  1. U H, or
  2. dim(UH) = dim(U) - 1.
In particular, any line not contained in H intersects H in precisely one point.

Note that we have defined projective planes in two ways, you should show that these two definitions coincide. (Homework problem #9)