Algebraic Geometry I

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.

"In the 20th century, algebraic geometry has undergone several revolutionary changes with respect to its conceptual foundations, technical framework, and intertwining with other branches of mathematics. Accordingly the way it is taught has gone through distinct phases. The theory of algebraic schemes, together with its full-blown machinery of sheaves and their cohomology, being for now the ultimate stage of this evolution process in algebraic geometry, had created -- around 1960 -- the urgent demand for new textbooks reflecting these developments and (henceforth) various facets of algebraic geometry. ... It was David Mumford, who at first started the project of writing a textbook on algebraic geometry in its new setting. His mimeographed Harvard notes ntroduction to algebraic geometry: Preliminary version of the first three chapters (bound in red) were distributed in the mid 1960's, and they were intended as the first stage of a forthcoming, more inclusive textbook. For some years, these mimeographed notes represented the almost only, however utmost convenient and abundant source for non-experts to get acquainted with the basic new concepts and ideas of modern algebraic geometry. Their timeless utility, in this regard, becomes apparent from the fact that two reprints of them have appeared, since 1988, as a proper book under the title he red book of varieties and schemes' ( Lect. Notes Math. 1358). In the process of exending his Harvard notes to a comprehensive textbook, the author's teaching experiences led him to the didactic conclusion that it would be better to split the book into two volumes, thereby starting with complex projective varieties (in volume I), and proceeding with schemes and their cohomology (in volume II). -- In 1976, the author published the first volume under the title lgebraic geometry. I: Complex projective varieties where the corrections concerned the wiping out of some misprints, inconsistent notations, and other slight inaccuracies. The book under review is an unchanged reprint of this corrected second edition from 1980. 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 MATH, 821

• 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 BirationalGeometry 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.