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Chapter Ⅸ．The Hilbert Scheme 1

1．Introduction 1

2．The idea of the Hilbert scheme 4

3．Flatness 12

4．Construction of the Hilbert scheme 19

5．The characteristic system 27

6．Mumford's example 40

7．Variants of the Hilbert scheme 43

8．Tangent space computations 49

9．Cn families of projective manifolds 56

10．Bibliographical notes and further reading 64

11．Exercises 65

Chapter Ⅹ．Nodal curves 79

1．Introduction 79

2．Elementary theory of nodal curves 83

3．Stable curves 99

4．Stable reduction 104

5．Isomorphisms of families of stable curves 113

6．The stable model,contraction,and projection 117

7．Clutching 126

8．Stabilization 127

9．Vanishing cycles and the Picard-Lefschetz transformation 143

10．Bibliographical notes and further reading 161

11．Exercises 161

Chapter Ⅺ．Elementary deformation theory and some applications 167

1．Introduction 167

2．Deformations of manifolds 172

3．Deformations of nodal curves 178

4．The concept of Kuranishi family 187

5．The Hilbert scheme of v-canonical curves 193

6．Construction of Kuranishi families 203

7．The Kuranishi family and continuous deformations 212

8．The period map and the local Torelli theorem 216

9．Curvature of the Hodge bundles 224

10．Deformations of symmetric products 242

11．Bibliographical notes and further reading 248

Chapter Ⅻ．The moduli space of stable curves 249

1．Introduction 249

2．Construction of moduli space as an analytic space 257

3．Moduli spaces as algebraic spaces 268

4．The moduli space of curves as an orbifold 274

5．The moduli space of curves as a stack,Ⅰ 279

6．The classical theory of descent for quasi-coherent sheaves 288

7．The moduli space of curves as a stack,Ⅱ 294

8．Deligne-Mumford stacks 299

9．Back to algebraic spaces 307

10．The universal curve,projections and clutchings 309

11．Bibliographical notes and further reading 323

12．Exercises 323

Chapter ⅩⅢ．Line bundles on moduli 329

1．Introduction 329

2．Line bundles on the moduli stack of stable curves 332

3．The tangent bundle to moduli and related constructions 344

4．The determinant of the cohomology and some applications 347

5．The Deligne pairing 366

6．The Picard group of moduli space 379

7．Mumford's formula 382

8．The Picard group of the hyperelliptic locus 387

9．Bibliographical notes and further reading 396

Chapter ⅩⅣ．Projectivity of the moduli space of stable curves 399

1．Introduction 399

2．A little invariant theory 400

3．The invariant-theoretic stability of linearly stable smooth curves 406

4．Numerical inequalities for families of stable curves 414

5．Projectivity of moduli spaces 425

6．Bibliographical notes and further reading 437

Chapter ⅩⅤ．The Teichmüller point of view 441

1．Introduction 441

2．Teichmüller space and the mapping class group 445

3．A little surface topology 453

4．Quadratic differentials and Teichmüller deformations 461

5．The geometry associated to a quadratic differential 472

6．The proof of Teichmüller's uniqueness theorem 479

7．Simple connectedness of the moduli stack of stable curves 483

8．Going to the boundary of Teichmüller space 485

9．Bibliographical notes and further reading 497

10．Exercises 498

Chapter ⅩⅥ．Smooth Galois covers of moduli spaces 501

1．Introduction 501

2．Level structures on smooth curves 508

3．Automorphisms of stable curves 515

4．Compactifying moduli of curves with level structure;a first attempt 518

5．Admissible G-covers 525

6．Automorphisms of admissible covers 536

7．Smooth covers of ？g 544

8．Totally unimodular lattices 551

9．Smooth covers of ？g,n 556

10．Bibliographical notes and further reading 562

11．Exercises 562

Chapter ⅩⅦ．Cycles in the moduli spaces of stable curves 565

1．Introduction 565

2．Algebraic cycles on quotients by finite groups 566

3．Tautological classes on moduli spaces of curves 570

4．Tautological relations and the tautological ring 573

5．Mumford's relations for the Hodge classes 585

6．Further considerations on cycles on moduli spaces 596

7．The Chow ring of ？o,p 599

8．Bibliographical notes and further reading 604

9．Exercises 605

Chapter ⅩⅤⅢ．Cellular decomposition of moduli spaces 609

1．Introduction 609

2．The arc system complex 613

3．Ribbon graphs 616

4．The idea behind the cellular decomposition of Mg,n 623

5．Uniformization 624

6．Hyperbolic geometry 627

7．The hyperbolic spine and the definition of ？ 636

8．The equivariant cellular decomposition of Teichmüller space 643

9．Stable ribbon graphs 648

10．Extending the cellular decomposition to a partial compactification of Teichmüller space 652

11．The continuity of ？ 655

12．Odds and ends 661

13．Bibliographical notes and further reading 665

Chapter ⅩⅨ．First consequences of the cellular decomposition 667

1．Introduction 667

2．The vanishing theorems for the rational homology of Mg,p 670

3．Comparing the cohomology of ？g,n to the one of its boundary strata 673

4．The second rational cohomology group of ？g,n 676

5．A quick overview of the stable rational cohomology of Mg,n and the computation of H1(Mg,n)and H2(Mg,n) 683

6．A closer look at the orbicell decomposition of moduli spaces 690

7．Combinatorial expression for the classes ψi 694

8．A volume computation 699

9．Bibliographical notes and further reading 708

10．Exercises 709

Chapter ⅩⅩ．Intersection theory of tautological classes 717

1．Introduction 717

2．Witten's generating series 721

3．Virasoro operators and the KdV hierarchy 726

4．The combinatorial identity 729

5．Feynman diagrams and matrix models 734

6．Kontsevich's matrix model and the equation L2Z=0 745

7．A nonvanishing theorem 750

8．A brief review of equivariant cohomology and the virtual Euler-Poincaré characteristic 754

9．The virtual Euler-Poincaré characteristic of Mg,n 759

10．A very quick tour of Gromov-Witten invariants 766

11．Bibliographical notes and further reading 771

12．Exercises 773

Chapter ⅩⅪ．Brill-Noether theory on a moving curve 779

1．Introduction 779

2．The relative Picard variety 781

3．Brill-Noether varieties on moving curves 788

4．Looijenga's vanishing theorem 796

5．The Zariski tangent spaces to the Brill-Noether varieties 802

6．The μl homomorphism 808

7．Lazarsfeld's proof of Petri's conjecture 814

8．The normal bundle and Horikawa's theory 819

9．Ramification 835

10．Plane curves 845

11．The Hurwitz scheme and its irreducibility 854

12．Plane curves and g l d's 863

13．Unirationality results 872

14．Bibliographical notes and further reading 879

15．Exercises 885

Bibliography 903

Index 945