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CHAPTER Ⅰ Preliminaries 1

1．Divisors and Line Bundles on Curves 1

2．The Riemann-Roch and Duality Theorems 6

3．Abel's Theorem 15

4．Abelian Varieties and the Theta Function 20

5．Poincaré's Formula and Riemann's Theorem 25

6．A Few Words About Moduli 28

Bibliographical Notes 30

Exercises 31

A．Elementary Exercises on Plane Curves 31

B．Projections 35

C．Ramification and Plücker Formulas 37

D．Miscellaneous Exercises on Linear Systems 40

E．Weierstrass Points 41

F．Automorphisms 44

G．Period Matrices 48

H．Elementary Properties of Abelian Varieties 48

APPENDIX A The Riemann-Roch Theorem,Hodge Theorem,and Adjoint Linear Systems 50

1．Applications of the Discussion About Plane Curves with Nodes 56

2．Adjoint Conditions in General 57

CHAPTER Ⅱ Determinantal Varieties 61

1．Tangent Cones to Analytic Spaces 61

2．Generic Determinantal Varieties:Geometric Description 67

3．The Ideal of a Generic Determinantal Variety 70

4．Determinantal Varieties and Porteous'Formula 83

(i)Sylvester's Determinant 87

(ii)The Top Chern Class of a Tensor Product 89

(iii)Porteous'Formula 90

(iv)What Has Been Proved 92

5．A Few Applications and Examples 93

Bibliographical Notes 100

Exercises 100

A．Symmetric Bilinear Maps 100

B．Quadrics 102

C．Applications of Porteous'Formula 104

D．Chern Numbers of Kernel Bundles 105

CHAPTER Ⅲ Introduction to Special Divisors 107

1．Clifford's Theorem and the General Position Theorem 107

2．Castelnuovo's Bound,Noether's Theorem,and Extremal Curves 113

3．The Enriques-Babbage Theorem and Petri's Analysis of the Canonical Ideal 123

Bibliographical Notes 135

Exercises 136

A．Symmetric Products of P1 136

B．Refinements of Clifford's Theorem 137

C．Complete Intersections 138

D．Projective Normality（Ⅰ） 140

E．Castelnuovo's Bound on k-Normality 141

F．Intersections of Quadrics 142

G．Space Curves of Maximum Genus 143

H．G.Gherardelli's Theorem 147

I．Extremal Curves 147

J．Nearly Castelnuovo Curves 149

K．Castelnuovo's Theorem 151

L．Secant Planes 152

CHAPTER Ⅳ The Varieties of Special Linear Series on a Curve 153

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

2．The Universal Divisor and the Poincaré Line Bundles 164

3．The Varieties W？（C）and G？（C）Parametrizing Special Linear Series on a Curve 176

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 Notes 195

Exercises 196

A．Elementary Exercises on μ0 196

B．An Interesting Identification 197

C．Tangent Spaces to W1(C) 197

D．Mumford's Theorem for g？'s 198

E．Martens-Mumford Theorem for Birational Morphisms 198

F．Linear Series on Some Complete Intersections 199

G．Keem's Theorems 200

CHAPTER Ⅴ The Basic Results of the Brill-Noether Theory 203

Bibliographical Notes 217

Exercises 218

A．W？（C）on a Curve C of Genus 6 218

B．Embeddings of Small Degree 220

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 Function 225

1．The Riemann Singularity Theorem 225

2．Kempf's Generalization of the Riemann Singularity Theorem 239

3．The Torelli Theorem 245

4．The Theory of Andreotti and Mayer 249

Bibliographical Notes 261

Exercises 262

A．The Difference Map φd（Ⅱ） 262

B．Refined Torelli Theorems 263

C．Translates of Wθ-1,Their Intersections,and the Torelli Theorem 265

D．Prill's Problem 268

E．Another Proof of the Torelli Theorem 268

F．Curves of Genus 5 270

G．Accola's Theorem 275

H．The Difference Map φd（Ⅲ） 276

I．Geometry of the Abelian Sum Map u in Low Genera 278

APPENDIX B Theta Characteristics 281

1．Norm Maps 281

2．The Weil Pairing 282

3．Theta Characteristics 287

4．Quadratic Forms Over Z/2 292

APPENDIX C Prym Varieties 295

Exercises 303

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

1．Ample Vector Bundles 304

2．The Existence Theorem 308

3．The Connectedness Theorem 311

4．The Class of W？（C） 316

5．The Class of C？ 321

Bibliographical Notes 326

Exercises 326

A．The Connectedness Theorem 326

B．Analytic Cohomology of Cd,d≤2g-2 328

C．Excess Linear Series 329

CHAPTER Ⅷ Enumerative Geometry of Curves 330

1．The Grothendieck-Riemann-Roch Formula 330

2．Three Applications of the Grothendieck-Riemann-Roch Formula 333

3．The Secant Plane Formula:Special Cases 340

4．The General Secant Plane Formula 345

5．Diagonals in the Symmetric Product 358

Bibliographical Notes 364

Exercises 364

A．Secant Planes to Canonical Curves 364

B．Weierstrass Pairs 365

C．Miscellany 366

D．Push-Pull Formulas for Symmetric Products 367

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

F．Every Curve Has a Base-Point-Free g？-1 372

Bibliography 375

Index 383