Projective Geometry - Lecture Notes Chapter 2

2.5 Normal Rational Curves

Def: Let P be a projective space of dimension d. We say that a set S of at least d+1 points of P is in general position if any d+1 points of S form a basis for P.

Examples:

  1. A set of at least 3 points of a projective plane are in general position iff no three are on a common line.
  2. A set of at least 4 points of a 3-dimensional projective space are in general position iff no 4 are in a common plane.
Def: Let P = P(V) be a d-dimensional projective space coordinatized over a field F. A normal rational curve is any set of points projectively equivalent to the set
C = {(1:t:t2:t3: ...:td) | t in F} {(0:0: ...:0:1)}.
Example: In PG(3,4) (see last section) the following set of points form a normal rational curve: (0:0:0:1), (1:0:0:0), (1:1:1:1), (1:a:a2:1) and (1:a2:a:1). Note that no 4 of these points lie in a common plane. This implies that no 3 of them lie on a common line.

Theorem 2.5.1: The points of a normal rational curve are in general position.

Corollary 2.5.2: Each hyperplane intersects a normal rational curve in at most d points.

2.6 The Moulton Plane

We now provide an example of a nondesarguesian plane by constructing an affine plane over the reals in which Desargues theorem does not hold universally. This example is known as the Moulton plane.

Def: Define the geometry M as follows:

Points
The pairs (x,y) with x,y .
Lines
Described by the equations of the form x = c and y = mx + b with c,m,b.
Incidence
(u,v) I (x=c) iff u = c. (u,v) I (y = mx + b) iff v = mu + b unless u < 0 and m < 0, in which case (u,v) I (y = mx + b) iff v = 2mu + b.
Theorem 2.6.1: The Moulton plane is an affine plane in which the theorem of Desargues is not true.

2.7 Spacial Geometries are Desarguesian

Theorem 2.7.1: Let P be a projective space of dimension d. If d 3, then the theorem of Desargues holds in P.

Corollary 2.7.2: Let P be a projective plane. The theorem of Desargues holds in P iff P can be embedded as a plane in a projective space of dimension 3.

Note: The converse of this corollary is also true, but requires more work to prove.