Basic algebraic geometry 1 : varieties in projective space基本的代数几何1
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作者： Igor R. Shafarevich著
出 版 社： 新世纪出版社
出版时间： 1994-8-1字数：版次： 1页数： 303印刷时间： 1994/08/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540548126包装： 平装内容简介
Basic Algebraic Geometry, Volume I, is a revised and expanded new edition of the first four chapters of Shafarevich's well-known introductory book on algebraic geometry. The author has added plenty of new, mostly concrete geometrical material such as Grassmannian varieties, plane cubic curves, the cubic surface, degenerations of quadrics and elliptic curves, the Bertini theorems, normal surface singularities. There are also some new number-theoretical applications. Shafarevich succeeds in making algebraic geometry accessible to non-specialists and beginners and his two-volume book will remain one of the most popular introductions to this field. The book is suitable for third-year undergraduates in mathematics and also for students of theoretical physics.
目录
BOOK 1. Varieties in projective space
Chapter I. Basic Notions
1. Algebraic Curves in the Plane
1.1. Plane Curves
1.2. Rational Curves
1.3. Relation with Field Theory
1.4. Rational Maps
1.5. Singular and Nonsingular Points
1.6. The Projective Plane
Exercises to 1
2. Closed Subsets of Affine Space
2.1. Definition of Closed Subsets
2.2. Regular Functions on a Closed Subset
2.3. Regular Maps
Exercises to 2
3. Rational Functions
3.1. Irreducible Algebraic Subsets
3.2. Rational Functions
3.3. Rational Maps
Exercises to 3
4. Quasiprojective Varieties
4.1. Closed Subsets of Projective Space
4.2. Regular Functions
4.3. Rational Functions
4.4. Examples of Regular Maps
Exercises to 4
5. Products and Maps of Quasiprojective Varieties
5.1. Products
5.2. The Image of a Projective Variety is Closed
5.3. Finite Maps
5.4. Noether Normalisation
Exercises to 5
6. Dimension
6.1. Definition of Dimension
6.2. Dimension of Intersection with a Hypersurface
6.3. The Theorem on the Dimension of Fibres
6.4. Lines on Surfaces
Exercises to 6
Chapter II. Local Properties
1. Singular and Nonsingular Points
1.1. The Local Ring of a Point
1.2. The Tangent Space
1.3. Intrinsic Nature of the Tangent Space
1.4. Singular Points
1.5. The Tangent Cone
Exercises to 1
2. Power Series Expansions
2.1. Local Parameters at a Point
2.2. Power Series Expansions
2.3. Varieties over the Reals and the Complexes
Exercises to 2
3. Properties of Nonsingular Points
3.1. Codimension 1 Subvarieties
3.2. Nonsingular Subvarieties
Exercises to 3
4. The Structure of Birational Maps
4.1. Blowup in Projective Space
4.2. Local Blowup
4.3. Behaviour of a Subvariety under a Blowup
4.4. Exceptional Subvarieties
4.5. Isomorphism and Birational Equivalence
Exercises to 4
5. Normal Varieties
5.1. Normal Varieties
5.2. Normalisation of an Affine Variety
5.3. Normalisation of a Curve
5.4. Projective Embedding of Nonsingular Varieties
Exercises to 5
6. Singularities of a Map
6.1. Irreducibility
6.2. Nonsingularity
6.3. Ramification
6.4. Examples
Exercises to 6
Chapter IIIDivisors and differential forms
Chapter iv:Intersecction numbers
Algebraic appendix
References
Index
BOOK 2. Schemes and varieties
Chapter v Schemes
Chapter vi Varieties
BOOK 3 Complex algebraic varieties and complex manifolds
Chapter vii The topology of algebraic varieties
Chapter viii Complex manifolds
Chapter IX Universal cover
Historical sketch
References
Index
