Algebraic Geometry

  1. Algebraic Curves
    1. Basic Properties of Curves
      1. Definition of Algebraic Curves
        1. Smooth and Singular Curves
          1. Examples of Curves
            1. Rational Curves
              1. Elliptic Curves
                1. Hyperelliptic Curves
              2. Function Fields of Curves
                1. Rational Function Field
                  1. Places and Valuations
                    1. Discrete Valuation Rings
                    2. Divisors on Curves
                      1. Effective Divisors
                        1. Degree of Divisors
                          1. Principal Divisors
                            1. Divisor Class Group
                            2. Riemann-Roch Theorem for Curves
                              1. Statement of the Theorem
                                1. Genus of a Curve
                                  1. Definition via Canonical Divisor
                                    1. Geometric Interpretation
                                      1. Computation Methods
                                      2. Canonical Divisor
                                        1. Definition and Properties
                                          1. Adjunction Formula
                                          2. Applications of Riemann-Roch
                                            1. Dimension of Linear Systems
                                              1. Existence of Morphisms to Projective Space
                                            2. Special Classes of Curves
                                              1. Rational Curves
                                                1. Characterization
                                                  1. Parametrizations
                                                  2. Elliptic Curves
                                                    1. Weierstrass Form
                                                      1. Group Law
                                                        1. Geometric Construction
                                                          1. Algebraic Formulation
                                                          2. j-Invariant
                                                            1. Isomorphism Classes
                                                            2. Hyperelliptic Curves
                                                              1. Definition and Properties
                                                                1. Canonical Embedding
                                                              2. Morphisms between Curves
                                                                1. Degree of Morphisms
                                                                  1. Ramification
                                                                    1. Ramification Points
                                                                      1. Ramification Index
                                                                      2. Hurwitz Formula
                                                                        1. Automorphisms of Curves
                                                                        2. Moduli of Curves
                                                                          1. Moduli Problems
                                                                            1. Moduli of Elliptic Curves
                                                                              1. Moduli of Higher Genus Curves
                                                                                1. Compactifications

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                                                                              10. Divisors and Line Bundles

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                                                                              12. Algebraic Surfaces