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Algebraic Geometry I: Complex Projective Varieties
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eBay item number:184052560500
Item specifics
- Condition
- Book Title
- Algebraic Geometry I: Complex Projective Varieties
- ISBN
- 9783540586579
- Publication Name
- Algebraic Geometry I : Complex Projective Varieties
- Item Length
- 9.3in
- Publisher
- Springer Berlin / Heidelberg
- Series
- Classics in Mathematics Ser.
- Publication Year
- 1995
- Type
- Textbook
- Format
- Trade Paperback
- Language
- English
- Features
- Reprint
- Item Width
- 6.1in
- Item Weight
- 22.6 Oz
- Number of Pages
- X, 186 Pages
About this product
Product Information
Let me begin with a little history. In the 20th century, algebraic geometry has gone through at least 3 distinct phases. In the period 1900-1930, largely under the leadership of the 3 Italians, Castelnuovo, Enriques and Severi, the subject grew immensely. In particular, what the late 19th century had done for curves, this period did for surfaces: a deep and systematic theory of surfaces was created. Moreover, the links between the "synthetic" or purely "algebro-geometric" techniques for studying surfaces, and the topological and analytic techniques were thoroughly explored. However the very diversity of tools available and the richness of the intuitively appealing geometric picture that was built up, led this school into short-cutting the fine details of all proofs and ignoring at times the time consuming analysis of special cases (e. g. , possibly degenerate configurations in a construction). This is the traditional difficulty of geometry, from High School Euclidean geometry on up. In the period 1930-1960, under the leadership of Zariski, Weil, and (towards the end) Grothendieck, an immense program was launched to introduce systematically the tools of commutative algebra into algebraic geometry and to find a common language in which to talk, for instance, of projective varieties over characteristic p fields as well as over the complex numbers. In fact, the goal, which really goes back to Kronecker, was to create a "geometry" incorporating at least formally arithmetic as well as projective geo metry.
Product Identifiers
Publisher
Springer Berlin / Heidelberg
ISBN-10
3540586571
ISBN-13
9783540586579
eBay Product ID (ePID)
465017
Product Key Features
Publication Name
Algebraic Geometry I : Complex Projective Varieties
Format
Trade Paperback
Language
English
Features
Reprint
Series
Classics in Mathematics Ser.
Publication Year
1995
Type
Textbook
Number of Pages
X, 186 Pages
Dimensions
Item Length
9.3in
Item Width
6.1in
Item Weight
22.6 Oz
Additional Product Features
Number of Volumes
1 Vol.
Lc Classification Number
Qa564-609
Edition Description
Reprint
Edition Number
2
Reviews
"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, "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
Table of Content
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.
Copyright Date
1995
Topic
Geometry / Algebraic
Lccn
94-039113
Dewey Decimal
516.3/5
Intended Audience
Scholarly & Professional
Dewey Edition
20
Illustrated
Yes
Genre
Mathematics
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