| Introduction Prerequisites Chapter 1. Affine Varieties 1A. Their Definition， Tangent Space， Dimension， Smooth and Singular Points. 1B. Analytic Uniformization at Smooth Points， Examples of Topological Knottedness at Singular Points 1C. Ox，xa UFD when x Smooth； Divisor of Zeroes and Poles of Functions Chapter 2. Projective Varieties 2A. Their Definition， Extension of Concepts from Aftine to Projective Case 2B. Products， Segre Embedding， Correspondences 2C. Elimination Theory， Noether's Normalization Lemma， Density of Zariski-Open Sets Chapter 3. Structure of Correspondences 3A. Local Properties——Smooth Maps， Fundamental Openness Principle， Zariski's Main Theorem 3B. Global Propcrties——Zariski's Connectedness Theorem， Specialization Principle 3C. Intersections on Smooth Varieties Chapter 4. Chow's Theorem 4A. Internally and Externally Defined Analytic Sets and their Local Descriptions as Branched Coverings of C'. 4B. Applications to Uniqueness of Algebraic Structure and Connectedness Chapter 5. Degree of a Projective Variety 5A. Definition of deg X， multxX， of the Blow up Bx（X）， Effect of a Projection， Examples 5B. Bezout's Theorem 5C. Volume of a Projective Variety； Review of Homology， DeRham's Theorem， Varieties as Minimal Submanifolds Chapter 6. Linear Systems 6A. The Correspondence between Linear Systems and Rational Maps， Examples； Complete Linear Systems are Finite-Dimensional 6B. Differential Forms， Canonical Divisors and Branch Loci 6C. Hilbert Polynomials， Relations with Degree Appendix to Chapter 6. The Weil-Samuel Algebraic Theory of Multiplicity Chapter 7. Curves and Their Genus 7A. Existence and Uniqueness of the Non-Singular Model of Each Function Field of Transcendence Degree 1 （after Albanese） 7B.Arithmetic Genus= Topological Genus； Existence of Good Projections to p1， p2， p3 7C. Residues of Differentials on Curves， the Classical Riemann-Roch Theorem for Curves and Applications 7D. Curves of Genus 1 as Plane Cubics and as Complex Tori C/L Chapter 8. The Birational Geometry of Surfaces 8A. Generalities on Blowing up Points 8B. Resolution of Singularities of Curves on a Smooth Surface by Blowing up the Surface； Examples 8C. Factorization of Birational Maps between Smooth Surfaces； the Trees of Infinitely Near Points 8D. The Birational Map between P“ and the Quadric and Cubic Surfaces； the 27 Lines on a Cubic Surface Bibliography List of Notations Index |
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