Real algebraic varieties are ubiquitous.They are the first objects encountered when learning of coordinates, then equations, but the systematic study of these objects, however elementary they may be, is formidable.

This book is intended for two kinds of audiences: it accompanies the reader, familiar with algebra and geometry at the masters level, in learning the basics of this rich theory, as much as it brings to the most advanced reader many fundamental results often missing from the available literature, the folklore. In particular, the introduction of topological methods of the theory to non-specialists is one of the original features of the book.

The first three chapters introduce the basis and classical methods of real and complex algebraic geometry. The last three chapters each focus on one more specific aspect of real algebraic varieties. A panorama of classical knowledgeis presented, as well as major developments of the last twenty years in the topology and geometry of varieties of dimension two and three, without forgetting curves, the central subject of Hilbert's famous sixteenth problem.

Various levels of exercises are given, and the solutions of many of them are provided at the end of each chapter.

The first book to introduce topological methods of the theory of real algebraic varieties to non-specialists Presents both classical results and recent developments in the topology and geometry of varieties of dimension two and three, without forgetting curves, the central subject of Hilbert's famous sixteenth problem Contains various levels of exercises with solutions to many of them provided at the end of each chapter

Autorentext
Frédéric Mangolte is Professor at Angers University in France. He is a specialist of real algebraic varieties, namely of their topology and their geometry. His research is focused on algebraic surfaces, algebraic threefolds and the Cremona group.

Klappentext

This book gives a systematic presentation of real algebraic varieties. Real algebraic varieties are ubiquitous.They are the first objects encountered when learning of coordinates, then equations, but the systematic study of these objects, however elementary they may be, is formidable. This book is intended for two kinds of audiences: it accompanies the reader, familiar with algebra and geometry at the masters level, in learning the basics of this rich theory, as much as it brings to the most advanced reader many fundamental results often missing from the available literature, the folklore . In particular, the introduction of topological methods of the theory to non-specialists is one of the original features of the book. The first three chapters introduce the basis and classical methods of real and complex algebraic geometry. The last three chapters each focus on one more specific aspect of real algebraic varieties. A panorama of classical knowledgeis presented, as well as major developments of the last twenty years in the topology and geometry of varieties of dimension two and three, without forgetting curves, the central subject of Hilbert's famous sixteenth problem. Various levels of exercises are given, and the solutions of many of them are provided at the end of each chapter.

Inhalt
Preface.- Introduction, Algebraic models of smooth manifolds.- 1. Algebraic varieties.- 2. R-varieties.- 3. Topology of varieties with an involution.- 4. Surfaces.- 5. Algebraic approximation.- 6. Three dimensional varieties.- Appendices.- Bibliography.- Glossary of Notations.- Index.- List of Examples.- List of Figures