Algebraic Geometry

  1. Algebraic Surfaces
    1. Basic Properties of Surfaces
      1. Definition and Examples
        1. Smooth Projective Surfaces
          1. Basic Invariants
            1. Geometric Genus
              1. Irregularity
                1. Picard Number
              2. Intersection Theory on Surfaces
                1. Intersection Numbers
                  1. Definition for Curves on Surfaces
                    1. Computation Methods
                    2. Intersection Form
                      1. Adjunction Formula
                        1. Statement and Applications
                          1. Genus Formula for Curves on Surfaces
                        2. Linear Systems on Surfaces
                          1. Complete Linear Systems
                            1. Base Points and Base Loci
                              1. Rational Maps from Linear Systems
                              2. Riemann-Roch for Surfaces
                                1. Statement of Riemann-Roch-Hirzebruch
                                  1. Chern Classes
                                    1. Applications to Line Bundles
                                      1. Kodaira Vanishing Theorem
                                      2. Special Classes of Surfaces
                                        1. Rational Surfaces
                                          1. Definition and Examples
                                            1. Birational Classification
                                            2. Ruled Surfaces
                                              1. Definition and Construction
                                                1. Properties and Examples
                                                2. K3 Surfaces
                                                  1. Definition and Basic Properties
                                                  2. Abelian Surfaces
                                                    1. Definition and Properties
                                                      1. Polarizations
                                                    2. Minimal Models and Classification
                                                      1. Minimal Models
                                                        1. Definition and Existence
                                                          1. Uniqueness Properties
                                                          2. Enriques-Kodaira Classification
                                                            1. Classification by Kodaira Dimension
                                                              1. Surface Types
                                                              2. Blow-ups and Contractions
                                                                1. Elementary Transformations
                                                                  1. Exceptional Curves

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                                                              11. Algebraic Curves

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                                                              13. Advanced Topics