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Introduction to Algebraic Geometry
About this book
This book presents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic $0$ and positive characteristic are emphasized. The basic tools of classical and modern algebraic geometry are introduced, including varieties, schemes, singularities, sheaves, sheaf cohomology, and intersection theory. Basic classical results on curves and surfaces are proved. More advanced topics such as ramification theory, Zariski's main theorem, and Bertini's theorems for general linear systems are presented, with proofs, in the final chapters.With more than 200 exercises, the book is an excellent resource for teaching and learning introductory algebraic geometry.
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Information
Table of contents
- Cover
- Title page
- Contents
- Preface
- Chapter 1. A Crash Course in Commutative Algebra
- Chapter 2. Affine Varieties
- Chapter 3. Projective Varieties
- Chapter 4. Regular and Rational Maps of Quasi-projective Varieties
- Chapter 5. Products
- Chapter 6. The Blow-up of an Ideal
- Chapter 7. Finite Maps of Quasi-projective Varieties
- Chapter 8. Dimension of Quasi-projective Algebraic Sets
- Chapter 9. Zariski’s Main Theorem
- Chapter 10. Nonsingularity
- Chapter 11. Sheaves
- Chapter 12. Applications to Regular and Rational Maps
- Chapter 13. Divisors
- Chapter 14. Differential Forms and the Canonical Divisor
- Chapter 15. Schemes
- Chapter 16. The Degree of a Projective Variety
- Chapter 17. Cohomology
- Chapter 18. Curves
- Chapter 19. An Introduction to Intersection Theory
- Chapter 20. Surfaces
- Chapter 21. Ramification and Étale Maps
- Chapter 22. Bertini’s Theorems and General Fibers of Maps
- Bibliography
- Index
- Back Cover