Projective geometry - Knowlpedia

Definition

Let $V$ be a vector space of dimension at least $2$ over a field $k$ . The projective geometry of $V$ is the incidence structure whose points are the one-dimensional linear subspaces of $V$ , and whose projective subspaces are the sets

\mathbb P(W)=\{L\subseteq W:\dim_kL=1\}

for nonzero linear subspaces $W\subseteq V$ . Incidence means containment.

Dimensions and joins

If $\dim_kW=r+1$ , then $\mathbb P(W)$ has projective dimension $r$ . In particular, projective lines are the $\mathbb P(W)$ with $\dim_kW=2$ , and projective hyperplanes arise from codimension-one subspaces of $V$ .

Two distinct points represented by lines $L_1,L_2\subset V$ lie on the unique projective line $\mathbb P(L_1+L_2)$ . More generally, sums and intersections of linear subspaces encode projective spans and projective intersections.

Coordinates and transformations

Choosing a basis identifies this incidence geometry with the $k$ -points of $\mathbb P_k^{n}$ , where $n=\dim_kV-1$ . Its linear symmetries form the projective general linear group. Allowing field automorphisms produces the larger projective semilinear group.

This construction is called Desarguesian projective geometry. Abstract projective planes need not arise from vector spaces over fields, so the vector-space hypothesis is substantive.

References

Emil Artin, Geometric Algebra, Interscience, 1957. Relevant: Chapter II, projective and affine geometry.
Peter J. Cameron, Projective and Polar Spaces, Queen Mary and Westfield College, 1992. Author-maintained text. Relevant: Chapter 1.