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Mathematics
Algebraic Geometry
1. Foundations of Algebraic Geometry
2. Polynomial Rings and Commutative Algebra
3. Gröbner Bases
4. Affine Algebraic Geometry
5. Projective Algebraic Geometry
6. Rational Maps and Birational Geometry
7. Local Properties and Singularities
8. Introduction to Schemes
9. Coherent Sheaves and Cohomology
10. Divisors and Line Bundles
11. Algebraic Curves
12. Algebraic Surfaces
13. Advanced Topics
10.
Divisors and Line Bundles
10.1.
Weil Divisors
10.1.1.
Definition on Normal Varieties
10.1.2.
Prime Divisors
10.1.3.
Divisor Group
10.1.4.
Principal Divisors
10.1.4.1.
Divisors of Rational Functions
10.1.5.
Linear Equivalence
10.1.6.
Divisor Class Group
10.2.
Cartier Divisors
10.2.1.
Definition via Local Equations
10.2.2.
Relationship to Weil Divisors
10.2.3.
Principal Cartier Divisors
10.2.4.
Cartier Divisor Group
10.3.
Line Bundles and Invertible Sheaves
10.3.1.
Definition of Line Bundles
10.3.2.
Invertible Sheaves
10.3.3.
Correspondence with Cartier Divisors
10.3.4.
Picard Group
10.3.4.1.
Definition and Properties
10.3.4.2.
Relationship to Divisor Class Group
10.4.
Linear Systems
10.4.1.
Definition and Examples
10.4.2.
Complete Linear Systems
10.4.3.
Base Locus
10.4.4.
Dimension of Linear Systems
10.5.
Ample and Very Ample Divisors
10.5.1.
Definitions and Characterizations
10.5.2.
Projective Embeddings
10.5.2.1.
Embeddings via Very Ample Line Bundles
10.5.3.
Ampleness Criteria
10.5.3.1.
Nakai-Moishezon Criterion
10.5.3.2.
Kleiman's Criterion
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9. Coherent Sheaves and Cohomology
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11. Algebraic Curves