Modern Projective Geometry / Edition 1

by Claude-Alain Faure, Alfred Frïlicher

Overview

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.

Product Details

ISBN-13:	9789048155446
Publisher:	Springer Netherlands
Publication date:	12/10/2010
Series:	Mathematics and Its Applications , #521
Edition description:	Softcover reprint of the original 1st ed. 2000
Pages:	363
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

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.

Editorial Reviews

Develops projective geometries and provides a systematic treatment of morphisms. This treatment is unique in that it does not confine itself to isomorphisms. Introduces a new fundamental theorem and its applications, describing homogeneous coordinates as morphisms of projective geometries by semilinear maps. Other topics include three equivalent definitions of projective geometries and their correspondence with certain lattices, quotients of projective geometries and isomorphism theorems, duality theory, and recent results in dimension theory. The authors are affiliated with the University of Geneva, Switzerland. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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