Plane Algebraic Curves

In a detailed and comprehensive introduction to the theory of plane algebraic curves, the authors examine this classical area of mathematics that both figured prominently in ancient Greek studies and remains a source of inspiration and topic of research to this day. Arising from notes for a course given at the University of Bonn in Germany, "Plane Algebraic Curves" reflects the author's concern for the student audience through emphasis upon motivation, development of imagination, and understanding of basic ideas. As classical objects, curves may be viewed from many angles; this text provides a foundation for the comprehension and exploration of modern work on singularities.

In the first chapter one finds many special curves with very attractive geometric presentations - the wealth of illustrations is a distinctive characteristic of this book - and an introduction to projective geometry (over the complex numbers). In the second chapter one finds a very simpleproof of Bezout's theorem and a detailed discussion of cubics. The heart of this book - and how else could it be with the first author - is the chapter on the resolution of singularities (always over the complex numbers). (...) Especially remarkable is the outlook to further work on the topics discussed, with numerous references to the literature. Many examples round off this successful representation of a classical and yet still very much alive subject.

(Mathematical Reviews)

Auteur

Egbert Brieskorn was a Professor of Mathematics at the University of Bonn, Germany.

Horst Knörrer is a Professor of Mathematics at the ETH Zurich, Switzerland.

Contenu
I. History of algebraic curves.- 1. Origin and generation of curves.- 2. Synthetic and analytic geometry.- 3. The development of projective geometry.- II. Investigation of curves by elementary algebraic methods.- 4. Polynomials.- 5. Definition and elementary properties of plane algebraic curves.- 6. The intersection of plane curves.- 7. Some simple types of curves.- III. Investigation of curves by resolution of singularities.- 8. Local investigations.- 9. Global investigations.- Bibliography.- Index.