Algebraic Geometry

  1. Projective Algebraic Geometry
    1. Motivation for Projective Space
      1. Limitations of Affine Geometry
        1. Points at Infinity
          1. Compactification
            1. Intersection Theory
            2. Projective Space
              1. Homogeneous Coordinates
                1. Definition and Equivalence
                  1. Coordinate Changes
                  2. Definition of Projective n-Space
                    1. Relationship to Affine Space
                      1. Examples and Visualizations
                      2. Homogeneous Polynomials
                        1. Definition and Degree
                          1. Homogenization and Dehomogenization
                            1. Properties under Coordinate Changes
                            2. Homogeneous Ideals
                              1. Definition and Examples
                                1. Graded Rings and Modules
                                  1. Irrelevant Ideal
                                  2. Projective Algebraic Sets
                                    1. Definition using Homogeneous Polynomials
                                      1. Well-definedness
                                        1. Relationship to Affine Algebraic Sets
                                        2. The Projective Nullstellensatz
                                          1. Statement and Proof
                                            1. Correspondence with Homogeneous Radical Ideals
                                            2. Zariski Topology on Projective Space
                                              1. Definition of Closed Sets
                                                1. Properties of the Topology
                                                  1. Irreducibility and Connectedness
                                                    1. Comparison with Affine Case
                                                    2. Affine Charts and Coordinate Patches
                                                      1. Covering Projective Space by Affine Opens
                                                        1. Coordinate Charts
                                                          1. Transition Maps between Charts
                                                            1. Gluing Construction
                                                            2. Projective Varieties
                                                              1. Definition and Examples
                                                                1. Irreducible Components
                                                                  1. Dimension of Projective Varieties
                                                                  2. The Homogeneous Coordinate Ring
                                                                    1. Definition and Properties
                                                                      1. Graded Structure
                                                                        1. Relationship to Projective Variety
                                                                        2. Morphisms of Projective Varieties
                                                                          1. Regular Functions on Projective Varieties
                                                                            1. Local Definition
                                                                              1. Global Regular Functions are Constant
                                                                              2. Morphisms between Projective Varieties
                                                                                1. Definition via Homogeneous Polynomials
                                                                              3. Products and Embeddings
                                                                                1. Products of Projective Spaces
                                                                                  1. Segre Embedding
                                                                                    1. Definition and Properties
                                                                                      1. Image and Equations
                                                                                      2. Veronese Embedding
                                                                                        1. Definition and Applications
                                                                                          1. Rational Normal Curves

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                                                                                      4. Affine Algebraic Geometry

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                                                                                      6. Rational Maps and Birational Geometry