Projective Geometry - Lecture Notes Chapter 3

The Fundamental Theorem of Projective Geometry

Now that we have determined the structure of essentially all projective geometries (projective planes still remain wildly different in their structures) we are in a position to classify the collineations (the bijections which preserve this structure). This classification is known as the Fundamental Theorem of Projective Geometry.

We will assume throughout this section that P is a Desarguesian projective space, so there exists a division ring F and vector space V over F so that P = P(V). Let H be a hyperplane of P and A = P\H be the affine space determined by H. Also, let O be a fixed point of A. It will be convenient to work in the affine space A. Recall that the points of A can be identified with the vectors in V and that the lines of A are the cosets of the 1-dimensional subspaces of V.

Let T = T(H) be the translation group of A, i.e., the group of all elations of P with axis H. Let G be the group of all collineations of A, and let GO be the subgroup of G consisting of all collineations of A which fix the point O. [The facts that G is a group and that GO is a subgroup of it are easy exercises.]

Lemma 3.5.1: (a) T is a normal subgroup of G.
(b) Each g in G can be uniquely written as g = tgO where t in T and gO in GO.

This lemma says that every collineation of A can be described as a product of a translation and an element of GO. Thus, we need only describe all the elements of these two subgroups in order to describe all collineations.

Lemma 3.5.2: Let t in T be an arbitrary translation. Choose an arbitrary point P of A and define P' = t(P). Viewing P and P' as vectors of V, we can describe t by:

t(X) = X + P' - P for all points X in A
We now turn our attention to GO.

Def: Let V be a vector space over the division ring F, and let be an automorphism of F. A map g of V into itself is called a semilinear map with companion automorphism if for all v, w in V and all a in F we have:

g(v+w) = g(v) + g(w), and
g(av) = (a)g(v).
If is the identity automorphism, then g is just a linear map.

Example:

Recall that in a finite field GF(q), with q = pe, the map x xp is an automorphism of the field. Let V be a 3-dimensional vector space over GF(2e). The map g: V V given by g (x,y,z) = (y2, z2, x2) is a semilinear map with companion automorphism being the squaring automorphism. To see this, let v = (x,y,z) and w = (x',y',z'), then g(v) =(y2,z2,x2) and g(w) = (y'2,z'2,x'2). Now, v + w = (x+x', y+y', z+z') and we have,

g(v+w) = ((y+y')2,(z+z')2,(x+x')2) = (y2 + y'2, z2 + z'2, x2+x'2) = g(v) + g(w), and
g(av) = ((ay)2, (az)2, (ax)2) = (a2y2, a2z2, a2x2) = a2(y2,z2,x2) = a2 g(v).

Lemma 3.5.3: Every element of GO is an additive map, that is, s in GO implies that s(v+w) = s(v) + s(w), for all v,w in V.

Theorem 3.5.4 : Every element of GO is a semilinear map of the vector space V.

We now turn to the projective space. It is easy to see that each collineation of A induces a collineation of P, but it is remarkable that the converse statement holds and this is the hard part of the following:

Theorem 3.5.8 : [The Fundamental Theorem of Projective Geometry] If P is a Desarguesian projective space and V a vector space such that P = P(V) then is a collineation of P if and only if there is a bijective semilinear map of V which induces .

We can now refine our view of the group of collineations of the projective space P, and see what role the central collineations play in this group.

Def: A collineation of the projective space P = P(V) is called a projective collineation (also known as a homography) if it is induced by a linear collineation of V (and so can be represented by a matrix with respect to some basis of V).

Theorem 3.6.1 : Every central collineation is a projective collineation.

Theorem 3.6.7 : Every projective collineation of P is a product of central collineations.

The group of collineations of P is denoted by PL(n,F) or PL(n,q) if F is a finite field of order q. The subgroup of homographies (projective collineations) is denoted by PGL(n,F) or PGL(n,q).