Introduction to Algebraic Geometry

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.

Table of Contents

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 Etale Maps

Chapter 22. Bertini’s Theorems and General Fibers of Maps