Opening a book on projective geometry, we expect an investigation of objects occurring in projective space. We expect to meet subspaces, quadrics, algebraic subvarieties, differential submanifolds, and many other objects. The book under review is not of this type, and this explains perhaps, why it carries the title Modern Projective Geometry. The main aim of the book is to introduce the category of projective geometries. This means that the authors’ goal is to look at projective geometries not only from inside, but also from outside. They adopt a synthetic definition of a projective geometry, and this definition has a fundamental influence on the style of the book. We find deep relations between projective geometries and other mathematical structures. First of all, a relation to lattice theory, i.e., the category of projective spaces is equivalent with the category of projective lattices. Secondly, a relation to closure spaces (and to matroids, in particular), i.e., an equivalence between the category of projective geometries and a category of certain closure spaces. The general approach leads us also to other geometries, e. g. affine geometries, hyperbolic geometries, and Möbius geometries. But this look at projective geometries from outside does not mean that we do not find information about these particular geometries. The book is written according to an excellent plan, and it surely represents a milestone in the development of projective geometry. The text is organized very carefully and each chapter is followed by many exercises. It is a great advantage of the book that it requires very modest prerequisites. Hence, it can be recommended already to undergraduate students in the first year of their study. On the other hand, I expect that also professional mathematicians will appreciate it.

Reviewer:

jiva