Projective Geometry - Lecture Notes Chapter 1

1.7 Diagrams

We now return to the concepts introduced in the first section and examine geometries from a new and more abstract perspective. We will be concerned only with geometries that have rank and will restrict our definitions to this case. The notions introduced apply in a more general setting and some of these can be found in the text.

Def: Let G be a geometry of rank r. The residue of a flag of G, denoted Res(), is the geometry consisting of all the elements of G which are incident with all the members of the flag with incidence induced from G. That is, an element of Res() is any element that can be added to the flag to produce a larger flag.

Note that since a flag can not contain two elements of the same type (Lemma 1.1.1), the elements of Res() can only be of types that are not present in . So, if G has rank r and the flag has s elements, then Res() is a geometry of rank r-s.

Examples:

  1. If is a maximal flag of the geometry G, then Res() is the empty set.
  2. Let G be a 4-dimensional projective space (i.e., a rank 4 projective geometry whose types are points, lines, planes and hyperplanes). Let be the flag {P,H} where P is a point and H a hyperplane containing P. Res() consists of all the lines and planes of G which contain P and are contained in H (thus incident with both P and H). Note that a line of Res() need not be incident with a plane of Res(), but they will intersect since both must contain P. Also notice that Res() is the same thing as H/P, the quotient geometry of H at P, and so, it is a projective plane.
  3. Same G as above. Let be the flag {P,} where P is a point and is a plane containing P. Res() consists of all the lines and hyperplanes of G which contain P and are contained in (lines) or contain (hyperplanes) the plane . Note that in this case, every line of Res() is incident with every hyperplane of Res().
  4. Again, with the same G as above, let be the flag {H} consisting of a hyperplane. Res() consists of all the points, lines and planes that are contained in H. Clearly, Res() is a 3-dimensional projective space (a rank 3 geometry).
  5. Let G be a 3-dimensional affine space and = {L,} where L is a line and an affine plane containing L. Res{) consists of all the points of L. Res() is a rank 1 geometry (there is only one type of object) and none of its elements are incident, i.e., this is nothing more than a set of points (any rank 1 geometry is just a set).

We are especially interested in residues which are rank 2 geometries. Each rank 2 geometry has a simple pictorial representation. This representation consists of two nodes and a labeled link joining them. Each of the nodes represents one of the two types of objects in the geometry while the label (or other characterization) of the link describes the incidence relation between them. We list below some of the more common rank 2 geometries and their representations.

Affine Plane
Projective Plane
This is so common that a label is omitted.
Trivial Geometry
The trivial geometry is one where every element of one type is incident with every element of the other type. If you like, you can think of the link here as being an invisible line.
Linear Space
A linear space is a rank 2 geometry in which every pair of points are on a unique line. Thus, projective planes and affine planes are linear spaces, but linear spaces can be more general than these - in particular, lines need not always contain the same number of points.
Generalized Quadrangle
If you don't know what a GQ is, ask Stan Payne.

Notice that the nodes in these representations are not labeled. Of course, they correspond to the points and blocks of the rank 2 geometry, but if the rank 2 geometry was the residue of a flag in some geometry G, the object types would also have names in G and these need not be the same, so a labeling could be confusing. In the examples of flag residues, (2) would be represented by and (3) would be represented by . The others do not have rank 2.

Def: Let G be a geometry of rank r. A diagram for G consists of r nodes where node i and node j are connected by a labeled link corresponding to the rank 2 geometry which describes the residues of all flags of G consisting of r-2 elements with type i and type j the missing ones.

Example: We will work out the diagram of an affine space of dimension 3. Since this is a rank 3 geometry, there will be 3 nodes which we can label points, lines and planes. To determine the link between the points node and the lines node, we look at all flags of size 1 (= 3-2) which do not contain points or lines, i.e., flags that consist of a single plane. The residues of all of these flags are of the same type, namely all the points and lines that are contained in the plane which is the flag. Since these planes are affine planes, the rank 2 geometry that describes them all is . Now consider the link between the lines node and the planes node. The flags we must look at consist of a single point. The residue of such a flag consists of all the lines and planes that contain this point. Call the point P. Since any two lines of the residue meet at P, they determine a unique plane that contains P. Any two of the affine planes that contain P must meet in a common line which contains P. Through any point of an affine plane there are at least 3 lines, so in any plane containing P there are at least 3 lines containing P. Thus, this residue has the structure of a projective plane. Since the structure of the residue did not depend on the point chosen as the flag, the link between the lines node and the planes node is . Finally, we examine the link between the points node and the planes node. The flags to look at are those consisting of a single line. The residue of such a flag consists of the points on that line and the planes containing that line. Since every point on the line is contained in every plane that contains this line, this residue is the trivial geometry. The link between the points node and the planes node is thus . We now put all this information together to get the diagram of our affine space of dimension 3, which is

Note that in the diagram of a geometry we do label the nodes.

Example: The diagram of a d-dimensional projective space is

and the diagram of a d-dimensional affine space is

Geometries which can be described by a diagram are known as Buekenhout-Tits geometries.

An interesting and active research question is: Given an arbitrarily formed diagram, is there a geometry which has this diagram?

Def: A geometry G is connected if for every two elements, x and y, of G there exists a sequence x = x0, x1, ..., xn = y of elements of G such that xi I xi+1 for i = 0,..., n-1.

Theorem 1.7.1: Each connected geometry with diagram

is a d-dimensional projective space.

Theorem 1.7.2: Each geometry of rank d with diagram

that contains a line with at least four points is a d-dimensional affine space.