代数曲线几何 第1卷 英文【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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CHAPTER Ⅰ Preliminaries1

1．Divisors and Line Bundles on Curves1

2．The Riemann-Roch and Duality Theorems6

3．Abel's Theorem15

4．Abelian Varieties and the Theta Function20

5．Poincaré's Formula and Riemann's Theorem25

6．A Few Words About Moduli28

Bibliographical Notes30

Exercises31

A．Elementary Exercises on Plane Curves31

B．Projections35

C．Ramification and Plücker Formulas37

D．Miscellaneous Exercises on Linear Systems40

E．Weierstrass Points41

F．Automorphisms44

G．Period Matrices48

H．Elementary Properties of Abelian Varieties48

APPENDIX A The Riemann-Roch Theorem,Hodge Theorem,and Adjoint Linear Systems50

1．Applications of the Discussion About Plane Curves with Nodes56

2．Adjoint Conditions in General57

CHAPTER Ⅱ Determinantal Varieties61

1．Tangent Cones to Analytic Spaces61

2．Generic Determinantal Varieties:Geometric Description67

3．The Ideal of a Generic Determinantal Variety70

4．Determinantal Varieties and Porteous'Formula83

(i)Sylvester's Determinant87

(ii)The Top Chern Class of a Tensor Product89

(iii)Porteous'Formula90

(iv)What Has Been Proved92

5．A Few Applications and Examples93

Bibliographical Notes100

Exercises100

A．Symmetric Bilinear Maps100

B．Quadrics102

C．Applications of Porteous'Formula104

D．Chern Numbers of Kernel Bundles105

CHAPTER Ⅲ Introduction to Special Divisors107

1．Clifford's Theorem and the General Position Theorem107

2．Castelnuovo's Bound,Noether's Theorem,and Extremal Curves113

3．The Enriques-Babbage Theorem and Petri's Analysis of the Canonical Ideal123

Bibliographical Notes135

Exercises136

A．Symmetric Products of P1136

B．Refinements of Clifford's Theorem137

C．Complete Intersections138

D．Projective Normality（Ⅰ）140

E．Castelnuovo's Bound on k-Normality141

F．Intersections of Quadrics142

G．Space Curves of Maximum Genus143

H．G.Gherardelli's Theorem147

I．Extremal Curves147

J．Nearly Castelnuovo Curves149

K．Castelnuovo's Theorem151

L．Secant Planes152

CHAPTER Ⅳ The Varieties of Special Linear Series on a Curve153

1．The Brill-Noether Matrix and the Variety C？154

2．The Universal Divisor and the Poincaré Line Bundles164

3．The Varieties W？（C）and G？（C）Parametrizing Special Linear Series on a Curve176

4．The Zariski Tangent Spaces to G？（C）and W？（C）185

5．First Consequences of the Infinitesimal Study of G？（C）and W？（C）191

Biographical Notes195

Exercises196

A．Elementary Exercises on μ0196

B．An Interesting Identification197

C．Tangent Spaces to W1(C)197

D．Mumford's Theorem for g？'s198

E．Martens-Mumford Theorem for Birational Morphisms198

F．Linear Series on Some Complete Intersections199

G．Keem's Theorems200

CHAPTER Ⅴ The Basic Results of the Brill-Noether Theory203

Bibliographical Notes217

Exercises218

A．W？（C）on a Curve C of Genus 6218

B．Embeddings of Small Degree220

C．Projective Normality（Ⅱ）221

D．The Difference Map φd:Cd×Cd→J(C)(I)223

CHAPTER Ⅵ The Geometric Theory of Riemann's Theta Function225

1．The Riemann Singularity Theorem225

2．Kempf's Generalization of the Riemann Singularity Theorem239

3．The Torelli Theorem245

4．The Theory of Andreotti and Mayer249

Bibliographical Notes261

Exercises262

A．The Difference Map φd（Ⅱ）262

B．Refined Torelli Theorems263

C．Translates of Wθ-1,Their Intersections,and the Torelli Theorem265

D．Prill's Problem268

E．Another Proof of the Torelli Theorem268

F．Curves of Genus 5270

G．Accola's Theorem275

H．The Difference Map φd（Ⅲ）276

I．Geometry of the Abelian Sum Map u in Low Genera278

APPENDIX B Theta Characteristics281

1．Norm Maps281

2．The Weil Pairing282

3．Theta Characteristics287

4．Quadratic Forms Over Z/2292

APPENDIX C Prym Varieties295

Exercises303

CHAPTER ⅦThe Existence and Connectedness Theorems for W？（C）304

1．Ample Vector Bundles304

2．The Existence Theorem308

3．The Connectedness Theorem311

4．The Class of W？（C）316

5．The Class of C？321

Bibliographical Notes326

Exercises326

A．The Connectedness Theorem326

B．Analytic Cohomology of Cd,d≤2g-2328

C．Excess Linear Series329

CHAPTER Ⅷ Enumerative Geometry of Curves330

1．The Grothendieck-Riemann-Roch Formula330

2．Three Applications of the Grothendieck-Riemann-Roch Formula333

3．The Secant Plane Formula:Special Cases340

4．The General Secant Plane Formula345

5．Diagonals in the Symmetric Product358

Bibliographical Notes364

Exercises364

A．Secant Planes to Canonical Curves364

B．Weierstrass Pairs365

C．Miscellany366

D．Push-Pull Formulas for Symmetric Products367

E．Reducibility of Wg-1 ⌒（Wg-1+u）（Ⅱ）370

F．Every Curve Has a Base-Point-Free g？-1372

Bibliography375

Index383