Modern Projective Geometry

Projective geometry is a very classical part of mathematics and one might think that the subject is completely explored and that there is nothing new to be added. But it seems that there exists no book on projective geometry which provides a systematic treatment of morphisms. We intend to fill this gap. It is in this sense that the present monograph can be called modern. The reason why morphisms have not been studied much earlier is probably the fact that they are in general partial maps between the point sets G and G, noted ' 9 : G -- ~ G', i.e. maps 9 : D -4 G' whose domain Dom 9 := D is a subset of G. We give two simple examples of partial maps which ought to be morphisms. The first example is purely geometric. Let E, F be complementary subspaces of a projective geometry G. If x E G \ E, then g(x) := (E V x) n F (where E V x is the subspace generated by E U {x}) is a unique point of F, i.e. one obtains a map 9 : G \ E -4 F. As special case, if E = {z} is a singleton and F a hyperplane with z tf. F, then g: G \ {z} -4 F is the projection with center z of G onto F.

Inhalt

1. Fundamental Notions of Lattice Theory.- 2. Projective Geometries and Projective Lattices.- 3. Closure Spaces and Matroids.- 4. Dimension Theory.- 5. Geometries of degree n.- 6. Morphisms of Projective Geometries.- 7. Embeddings and Quotient-Maps.- 8. Endomorphisms and the Desargues Property.- 9. Homogeneous Coordinates.- 10. Morphisms and Semilinear Maps.- 11. Duality.- 12. Related Categories.- 13. Lattices of Closed Subspaces.- 14. Orthogonality.- List of Problems.- List of Axioms.- List of Symbols.