Algebraic Geometry I

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,X a UFD when x Smooth; Divisor of Zeroes and Poles of Functions.- 2. Projective Varieties.-
2A. Their Definition, Extension of Concepts from Affine to Projective Case.-
2B. Products, Segre Embedding, Correspondences.-
2C. Elimination Theory, Noether's Normalization Lemma, Density of Zariski-Open Sets.- 3. Structure of Correspondences.-
3A. Local Properties-Smooth Maps, Fundamental Openness Principle, Zariski's Main Theorem.-
3B. Global Properties-Zariski's Connectedness Theorem, Specialization Principle.-
3C. Intersections on Smooth Varieties.- 4. Chow's Theorem.-
4A. Internally and Externally Defined Analytic Sets and their Local Descriptions as Branched Coverings of ?n.-
4B. Applications to Uniqueness of Algebraic Structure and Connectedness.- 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.- 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.- 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 ?1, ?2, ?3.-
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 ?/L.- 8. The Birational

Biography of David Mumford

David Mumford was born on June 11, 1937 in England and has been associated with Harvard University continuously from entering as freshman to his present position of Higgins Professor of Mathematics.

Mumford worked in the fields of Algebraic Gemetry in the 60's and 70's, concentrating especially on the theory of moduli spaces: spaces which classify all objects of some type, such as all curves of a given genus or all vector bundles on a fixed curve of given rank and degree. Mumford was awarded the Fields Medal in 1974 for his work on moduli spaces and algebraic surfaces. He is presently working on the mathematics of pattern recognition and artificial intelligence.