Description
Springer Algebraic Geometry I Complex Projective Varieties by David Mumford
From the reviews: "Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford's "Volume I" is, together with its predecessor the red book of varieties and schemes, now as before one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!" Zentralblatt_x000D_ Table of contents :- _x000D_
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 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 ?2 and the Quadric and Cubic Surfaces; the 27 Lines on a Cubic Surface.- List of Notations._x000D_