基础代数几何 第1卷 第3版 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

基础代数几何 第1卷 第3版 英文版PDF格式电子书版下载

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Book 1：Varieties in Projective Space3

1 Basic Notions3

1 Algebraic Curves in the Plane3

1.1 Plane Curves3

1.2 Rational Curves6

1.3 Relation with Field Theory9

1.4 Rational Maps11

1.5 Singular and Nonsingular Points13

1.6 The Projective Plane17

1.7 Exercises to Section 122

2 Closed Subsets of Affine Space23

2.1 Definition of Closed Subsets23

2.2 Regular Functions on a Closed Subset25

2.3 Regular Maps27

2.4 Exercises to Section 232

3 Rational Functions34

3.1 Irreducible Algebraic Subsets34

3.2 Rational Functions36

3.3 Rational Maps37

3.4 Exercises to Section 340

4 Quasiprojective Varieties41

4.1 Closed Subsets of Projective Space41

4.2 Regular Functions46

4.3 Rational Functions50

4.4 Examples of Regular Maps52

4.5 Exercises to Section 453

5 Products and Maps of Quasiprojective Varieties54

5.1 Products54

5.2 The Image of a Projective Variety is Closed57

5.3 Finite Maps60

5.4 Noether Normalisation65

5.5 Exercises to Section 565

6 Dimension66

6.1 Definition of Dimension66

6.2 Dimension of Intersection with a Hypersurface69

6.3 The Theorem on the Dimension of Fibres75

6.4 Lines on Surfaces77

6.5 Exercises to Section 680

2 Local Properties83

1 Singular and Nonsingular Points83

1.1 The Local Ring of a Point83

1.2 The Tangent Space85

1.3 Intrinsic Nature of the Tangent Space86

1.4 Singular Points92

1.5 The Tangent Cone94

1.6 Exercises to Section 195

2 Power Series Expansions98

2.1 Local Parameters at a Point98

2.2 Power Series Expansions100

2.3 Varieties over the Reals and the Complexes104

2.4 Exercises to Section 2106

3 Properties of Nonsingular Points106

3.1 Codimension 1 Subvarieties106

3.2 Nonsingular Subvarieties110

3.3 Exercises to Section 3111

4 The Structure of Birational Maps113

4.1 Blowup in Proiective Space113

4.2 Local Blowup115

4.3 Behaviour of a Subvariety Under a Blowup117

4.4 Exceptional Subvarieties119

4.5 Isomorphism and Birational Equivalence120

4.6 Exercises to Section 4123

5 Normal Varieties124

5.1 Normal Varieties124

5.2 Normalisation of an Affine Variety128

5.3 Normalisation of a Curve130

5.4 Projective Embedding of Nonsingular Varieties134

5.5 Exercises to Section 5136

6 Singularities of a Map137

6.1 Irreducibility137

6.2 Nonsingularity139

6.3 Ramification140

6.4 Examples143

6.5 Exercises to Section 6146

3 Divisors and Differential Forms147

1 Divisors147

1.1 The Divisor of a Function147

1.2 Locally Principal Divisors151

1.3 Moving the Support of a Divisor away from a Point153

1.4 Divisors and Rational Maps155

1.5 The Linear System of a Divisor156

1.6 Pencil of Conics over P1159

1.7 Exercises to Section 1161

2 Divisors on Curves163

2.1 The Degree ofa Divisor on a Curve163

2.2 Bézout's Theorem on a Curve167

2.3 The Dimension of a Divisor168

2.4 Exercises to Section 2169

3 The Plane Cubic170

3.1 The Class Group170

3.2 The Group Law173

3.3 Maps177

3.4 Applications179

3.5 Algebraically Nonclosed Field181

3.6 Exercises to Section 3183

4 Algebraic Groups184

4.1 Algebraic Groups184

4.2 Quotient Groups and Chevalley's Theorem185

4.3 Abelian Varieties186

4.4 The Picard Variety188

4.5 Exercises to Section 4189

5 Differential Forms190

5.1 Regular Differential 1-Forms190

5.2 Algebraic Definition of the Module of Differentials193

5.3 Differential p-Forms195

5.4 Rational Difierential Forms197

5.5 Exercises to Section5199

6 Examples and Applications of Differential Forms200

6.1 Behaviour Under Maps200

6.2 Invariant Differential Forms on a Group202

6.3 The Canonical Class204

6.4 Hypersurfaces206

6.5 Hyperelliptic Curves209

7 The Riemann-Roch Theorem on Curyes210

7.1 Statement of the Theorem210

7.2 Preliminary Form of the Riemann-Roch Theorem213

7.3 The Residue ofa 1-Form217

7.4 Linear Algebra in Infinite Dimensional Vector Spaces219

7.5 The Residue Theorem224

7.6 The Duality Theorem225

7.7 Exercises to Sections 6-7227

8 Higher Dimensional Generalisations229

4 Intersection Numbers233

1 Definition and Basic Properties233

1.1 Definition of Intersection Number233

1.2 Additivity236

1.3 Invariance Under Linear Equivalence238

1.4 The General Definition of Intersection Number242

1.5 Exercises to Section 1245

2 Applications of Intersection Numbers246

2.1 Bézout's Theorem in Projective and Multiprojective Space246

2.2 Varieties over the Reals248

2.3 The Genus of aN onsingular Curve on a Surface251

2.4 The Riemann-Roch Inequality on a Surface253

2.5 The Nonsingular Cubic Surface255

2.6 The Ring ofCycle Classes258

2.7 Exercises to Section 2259

3 Birational Maps of Surfaces260

3.1 Blowups of Surfaces260

3.2 Some Intersection Numbers261

3.3 Resolution of Indeterminacy263

3.4 Factorisation as a Chain of Blowups264

3.5 Remarks and Examples267

3.6 Exercises to Section3269

4 Singularities270

4.1 Singular Points of a Curve270

4.2 Surface Singularities273

4.3 Du Val Singularities274

4.4 Degeneration of Curves278

4.5 Exercises to Section4281

Algebraic Appendix283

1 Linear and Bilinear Algebra283

2 Polynomials285

3 Quasilinear Maps285

4 Invariants287

5 Fields288

6 Commutative Rings289

7 Unique Factorisation292

8 Integral Elements293

9 Length of a Module294

References297

Index301

Book 2：Schemes and Varieties3

5 Schemes3

1 The Spec of a Ring5

1.1 Definition of Spec A5

1.2 Properties of Points of Spec A7

1.3 The Zariski Topology of Spec A9

1.4 Irreducibility,Dimension11

1.5 Exercises to Section 114

2 Sheaves15

2.1 Presheaves15

2.2 The Structure Presheaf17

2.3 Sheaves19

2.4 Stalks of a Sheaf23

2.5 Exercises to Section 224

3 Schemes25

3.1 Definition of a Scheme25

3.2 Glueing Schemes30

3.3 Closed Subschemes32

3.4 Reduced Schemes and Nilpotents35

3.5 Finiteness Conditions36

3.6 Exercises to Section 338

4 Products of Schemes40

4.1 Definition of Product40

4.2 Group Schemes42

4.3 Separatedness43

4.4 Exercises to Section 446

6 Varieties49

1 Definitions and Examples49

1.1 Definitions49

1.2 Vector Bundles53

1.3 Vector Bundles and Sheaves56

1.4 Divisors and Line Bundles63

1.5 Exercises to Section 167

2 Abstract and Quasiprojective Varieties68

2.1 Chow's Lemma68

2.2 Blowup Along a Subvariety70

2.3 Example of Non-quasiprojective Variety74

2.4 Criterions for Projectivity79

2.5 Exercises to Section 281

3 Coherent Sheaves81

3.1 Sheaves of Ox-Modules81

3.2 Coherent Sheaves85

3.3 Dévissage of Coherent Sheaves88

3.4 The Finiteness Theorem92

3.5 Exercises to Section 393

4 Classification of Geometric Obiects and Universal Schemes94

4.1 Schemes and Functors94

4.2 The Hilbert Polynomial100

4.3 Flat Families103

4.4 The Hilbert Scheme107

4.5 Exercises to Section 4110

Book 3：Complex Algebraic Varieties and Complex Manifolds110

7 The Topology of Algebraic Varieties115

1 The Complex Topology115

1.1 Definitions115

1.2 Algebraic Varieties as Differentiable Manifolds；Orientation117

1.3 Homology of Nonsingular Projective Varieties118

1.4 Exercises to Section 1121

2 Connectedness121

2.1 Preliminary Lemmas121

2.2 The First Proof of the Main Theorem122

2.3 The Second Proof124

2.4 Analytic Lemmas126

2.5 Connectedness of Fibres127

2.6 Exercises to Section 2128

3 The Topology of Algebraic Curves129

3.1 Local Structure of Morphisms129

3.2 Triangulation of Curves131

3.3 Topological Classification of Curves133

3.4 Combinatorial Classification of Surfaces137

3.5 The Topology of Singularities of Plane Curves140

3.6 Exercises to Section 3142

4 Real Algebraic Curves142

4.1 Complex Conjugation143

4.2 Proof of Harnack's Theorem144

4.3 Ovals of Real Curves146

4.4 Exercises to Section 4147

8 Complex Manifolds149

1 Definitions and Examples149

1.1 Definition149

1.2 Quotient Spaces152

1.3 Commutative Algebraic Groups as Quotient Spaces155

1.4 Examples of Compact Complex Manifolds not Isomorphic to Algebraic Varieties157

1.5 Complex Spaces163

1.6 Exercises to Section 1165

2 Divisors and Meromorphic Functions166

2.1 Divisors166

2.2 Meromorphic Functions169

2.3 The Structure of the Field M（X）171

2.4 Exercises to Section 2174

3 Algebraic Varieties and Complex Manifolds175

3.1 Comparison Theorems175

3.2 Example of Nonisomorphic Algebraic Varieties that Are Isomorphic as Complex Manifolds178

3.3 Example of a Nonalgebraic Compact Complex Manifold with Maximal Number of Independent Meromorphic Functions181

3.4 The Classification of Compact Complex Surfaces183

3.5 Exercises to Section 3185

4 K？hler Manifolds185

4.1 K？hler Metric186

4.2 Examples188

4.3 Other Characterisations of K？hler Metrics190

4.4 Applications of K？hler Metrics193

4.5 Hodge Theory196

4.6 Exercises to Section 4198

9 Uniformisation201

1 The Universal Cover201

1.1 The Universal Cover of a Complex Manifold201

1.2 Universal Covers of Algebraic Curves203

1.3 Projective Embedding of Quotient Spaces205

1.4 Exercises to Section 1206

2 Curves of Parabolic Type207

2.1 Theta Functions207

2.2 Projective Embedding209

2.3 Elliptic Functions，Elliptic Curves and Elliptic Integrals210

2.4 Exercises to Section 2213

3 Curves of Hyperbolic Type213

3.1 Poincaré Series213

3.2 Projective Embedding216

3.3 Algebraic Curves and Automorphic Functions218

3.4 Exercises to Section 3221

4 Uniformising Higher Dimensional Varieties221

4.1 Complete Intersections are Simply Connected221

4.2 Example of Manifold with π1 a Given Finite Group222

4.3 Remarks226

4.4 Exercises to Section 4227

Historical Sketch229

1 Elliptic Integrals229

2 Elliptic Functions231

3 Abelian Integrals233

4 Riemann Surfaces235

5 The Inversion of Abelian Integrals237

6 The Geometry of Algebraic Curves239

7 Higher Dimensional Geometry241

8 The Analytic Theory of Complex Manifolds243

9 Algebraic Varieties over Arbitrary Fields and Schemes244

References247

References for the Historical Sketch250

Index253