Projective Geometry - Lecture Notes Chapter 1

1. The points of A are the points of P\.
2. The lines of A are the lines of P not in .
3. The t-dimensional subspaces of A are the t-dimensional subspaces of P which are not contained in .
4. Incidence is inherited.

For fixed t (1td-1), the rank 2 geometry consisting of the points and t-dimensional subspaces of A is denoted by A_t (thus affine space is the geometry A₁).

Subspaces of an affine space are called flats.

is called the hyperplane at infinity and its points are called points at infinity (sometimes improper points).

P is called the projective closure of A.

Examples:

1.

2.

3. Consider the example of section 1.4. Let be the plane 0001^T. The points of this plane (points at infinity) are those with last coordinate 0. The points of the affine space P\ are thus: 0001, 0011, 0101, 0111, 1001, 1011, 1101, and 1111. The projective plane 1000^T becomes the affine plane whose points are : 0001, 0011, 0101 and 0111. The line labeled a of the projective space becomes the affine line whose points are 0001 and 1001.

Def: Let G = (,,I) be a rank 2 geometry. A parallelism of G is an equivalence relation || on the block set satisfying Playfair's axiom (i.e., given a point P and a block B not containing P, there is a unique block B' containing P with B'||B.)

Theorem 1.6.1: For t {1,...,d-1}, A_t has a parallelism.

The parallelism of the above theorem is called the natural parallelism.

For the natural parallelism, any two distinct parallel t-flats span a (t+1)-flat. Two subspaces of arbitrary dimension are parallel if one is parallel to a subspace of the other. Thus, if is the hyperplane at infinity of P, then subspaces U and W are parallel if U W, or vice-versa. In particular, a line g is parallel to a hyperplane H if g is parallel to some line h of H, meaning that gh.

Lemma 1.6.2: Let A = P\ where P is a d-dimensional projective space.

1. Each line not parallel to a hyperplane H meets H in precisely one point of A.
2. If d = 2, any two non-parallel lines intersect in a point of A.

1. Through any two distinct points there passes exactly one line.
2. Playfair's axiom.
3. There exist 3 points not on a common line.

Theorem 1.6.5: Let A be a finite d-dimensional affine space of order q. Then

1. There exists an integer q2 such that any line of A is incident with exactly q points.
2. If U is a t-flat, then |U| = q^t.

Theorem 1.6.7: If S is a rank 2 geometry satisfying:

1. 2 points determine a unique line,
2. there are q² points in total, and
3. each line has exactly q points.