The purpose of this book is to revive some of the beautiful results obtained by various geometers of the 19th century, and to give its readers a taste of concrete algebraic geometry. A good deal of space is devoted to cross-ratios, conics, quadrics, and various interesting curves and surfaces. The fundamentals of projective geometry are efficiently dealt with by using a modest amount of linear algebra. An axiomatic characterization of projective planes is also given. While the topology of projective spaces over real and complex fields is described, and while the geometry of the complex projective libe is applied to the study of circles and Möbius transformations, the book is not restricted to these fields. Interesting properties of projective spaces, conics, and quadrics over finite fields are also given. This book is the first volume in the Readings in Mathematics sub-series of the UTM. From the reviews: "...The book of P. Samuel thus fills a gap in the literature. It is a little jewel. Starting from a minimal background in algebra, he succeeds in 160 pages in giving a coherent exposition of all of projective geometry. ... one reads this book like a novel. " D.Lazard in Gazette des Mathématiciens#1
1. Projective Spaces.- 1.1 Projective Spaces and Projective Bases.- 1.2 Projective Transformations and the Projective Group.- 1.3 Projective and Affine Spaces.- 1.4 Axiomatic Presentation of Projective and Affine Planes.- 1.5 Projective Spaces of Hyperplanes and Duality.- 1.6 The Projective Space of Circles.- 1.7 The Projective Space of Conics.- 1.8 Projective Spaces of Divisors in Algebraic Geometry.- 2. One-Dimensional Projective Geometry.- 2.1 Cross-ratios and Rational Maps.- 2.2 Cross-ratios and permutations.- 2.3 Harmonic Division.- 2.4 Projective Transformations and Involutions on a Projective Line.- 2.5 The Projective Structure of a Conic.- 2.6 Unicursal Curves.- 2.7 The Complex Projective Line and the Circular Group.- 2.8 Topology of Projective Spaces.- 3. Classification of Conics and Quadrics.- 3.1 What Is a Quadric?.- 3.2 Classification of Affine and Euclidean Quadrics.- 3.3 Projective Classification of Real Quadrics.- 3.4 Classification of Conies and Quadrics over a Finite Field.- 4. Polarity with Respect to a Quadric.- 4.1 Polars and Poles.- 4.2 Polarity with Respect to Conics.- 4.3 Polarity and Tangential Equations.- 4.4 Applications to Conics.- Appendix: (2,2)-Correspondences.- Index of Symbols and Notations.