This volume studies geometric objects defined by polynomial equations. It connects commutative algebra, geometry, and number theory through the language of varieties and schemes.
Part I. Affine Algebraic Geometry
Chapter 1. Affine Varieties
1.1 Polynomial rings and ideals 1.2 Zero sets of polynomials 1.3 Affine space 1.4 Zariski topology 1.5 Examples
Chapter 2. Coordinate Rings
2.1 Functions on varieties 2.2 Radical ideals 2.3 Nullstellensatz 2.4 Correspondence between ideals and varieties 2.5 Morphisms of varieties
Chapter 3. Algebra–Geometry Correspondence
3.1 Geometric interpretation of algebra 3.2 Irreducibility 3.3 Dimension 3.4 Regular functions 3.5 Examples
Part II. Projective Geometry
Chapter 4. Projective Space
4.1 Homogeneous coordinates 4.2 Projective varieties 4.3 Embeddings 4.4 Hyperplanes 4.5 Examples
Chapter 5. Projective Varieties
5.1 Homogeneous ideals 5.2 Projective Nullstellensatz 5.3 Morphisms 5.4 Rational maps 5.5 Intersection theory basics
Chapter 6. Curves and Surfaces
6.1 Algebraic curves 6.2 Singularities 6.3 Genus (overview) 6.4 Surfaces 6.5 Examples
Part III. Schemes
Chapter 7. Spec of a Ring
7.1 Prime spectrum 7.2 Zariski topology on Spec 7.3 Structure sheaf 7.4 Affine schemes 7.5 Examples
Chapter 8. General Schemes
8.1 Gluing affine schemes 8.2 Morphisms of schemes 8.3 Fiber products 8.4 Base change 8.5 Examples
Chapter 9. Properties of Schemes
9.1 Reduced and irreducible schemes 9.2 Noetherian schemes 9.3 Dimension 9.4 Separated and proper morphisms 9.5 Applications
Part IV. Sheaves and Cohomology
Chapter 10. Sheaves
10.1 Presheaves and sheaves 10.2 Sheaf operations 10.3 Stalks 10.4 Examples 10.5 Applications
Chapter 11. Cohomology
11.1 Derived functors (overview) 11.2 Čech cohomology 11.3 Sheaf cohomology 11.4 Vanishing theorems (overview) 11.5 Applications
Chapter 12. Line Bundles and Divisors
12.1 Cartier and Weil divisors 12.2 Line bundles 12.3 Picard group 12.4 Linear systems 12.5 Applications
Part V. Advanced Topics
Chapter 13. Intersection Theory
13.1 Intersection multiplicities 13.2 Chow groups (overview) 13.3 Basic constructions 13.4 Applications 13.5 Examples
Chapter 14. Singularities
14.1 Types of singularities 14.2 Resolution of singularities (overview) 14.3 Normalization 14.4 Local properties 14.5 Applications
Chapter 15. Moduli Spaces
15.1 Parameter spaces 15.2 Families of varieties 15.3 Representability 15.4 Examples 15.5 Applications
Part VI. Arithmetic and Geometric Connections
Chapter 16. Varieties over Fields
16.1 Varieties over finite fields 16.2 Rational points 16.3 Zeta functions (overview) 16.4 Applications 16.5 Examples
Chapter 17. Arithmetic Geometry
17.1 Schemes over integers 17.2 Diophantine geometry 17.3 Elliptic curves 17.4 Heights (overview) 17.5 Applications
Chapter 18. Complex Algebraic Geometry
18.1 Varieties over complex numbers 18.2 Analytic spaces 18.3 Hodge theory (overview) 18.4 Topological invariants 18.5 Applications
Part VII. Computational and Applied Aspects
Chapter 19. Computational Algebraic Geometry
19.1 Gröbner bases revisited 19.2 Elimination theory 19.3 Solving polynomial systems 19.4 Complexity issues 19.5 Software tools
Chapter 20. Applications
20.1 Coding theory 20.2 Cryptography 20.3 Robotics and kinematics 20.4 Computer vision 20.5 Data science
Chapter 21. Tropical Geometry (Overview)
21.1 Tropical semiring 21.2 Tropical varieties 21.3 Combinatorial structures 21.4 Applications 21.5 Connections
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Derived algebraic geometry (overview) 22.2 Stacks (overview) 22.3 Mirror symmetry (overview) 22.4 Noncommutative geometry (overview) 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Classification of varieties 23.2 Singularities and resolution 23.3 Moduli questions 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of algebraic geometry 24.2 Key contributors 24.3 Evolution from classical to modern 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Standard constructions reference B. Key theorems summary C. Proof templates D. Computational tools E. Cross-reference to other MSC branches
This volume builds algebraic geometry from classical varieties to modern scheme theory. It emphasizes the algebra–geometry correspondence and the structural role of commutative algebra.