代数曲线几何 第2卷 第2分册PDF|Epub|txt|kindle电子书版本网盘下载

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Chapter Ⅸ．The Hilbert Scheme1

1．Introduction1

2．The idea of the Hilbert scheme4

3．Flatness12

4．Construction of the Hilbert scheme19

5．The characteristic system27

6．Mumford's example40

7．Variants of the Hilbert scheme43

8．Tangent space computations49

9．Cn families of projective manifolds56

10．Bibliographical notes and further reading64

11．Exercises65

Chapter Ⅹ．Nodal curves79

1．Introduction79

2．Elementary theory of nodal curves83

3．Stable curves99

4．Stable reduction104

5．Isomorphisms of families of stable curves113

6．The stable model,contraction,and projection117

7．Clutching126

8．Stabilization127

9．Vanishing cycles and the Picard-Lefschetz transformation143

10．Bibliographical notes and further reading161

11．Exercises161

Chapter Ⅺ．Elementary deformation theory and some applications167

1．Introduction167

2．Deformations of manifolds172

3．Deformations of nodal curves178

4．The concept of Kuranishi family187

5．The Hilbert scheme of v-canonical curves193

6．Construction of Kuranishi families203

7．The Kuranishi family and continuous deformations212

8．The period map and the local Torelli theorem216

9．Curvature of the Hodge bundles224

10．Deformations of symmetric products242

11．Bibliographical notes and further reading248

Chapter Ⅻ．The moduli space of stable curves249

1．Introduction249

2．Construction of moduli space as an analytic space257

3．Moduli spaces as algebraic spaces268

4．The moduli space of curves as an orbifold274

5．The moduli space of curves as a stack，Ⅰ279

6．The classical theory of descent for quasi-coherent sheaves288

7．The moduli space of curves as a stack，Ⅱ294

8．Deligne-Mumford stacks299

9．Back to algebraic spaces307

10．The universal curve,projections and clutchings309

11．Bibliographical notes and further reading323

12．Exercises323

Chapter ⅩⅢ．Line bundles on moduli329

1．Introduction329

2．Line bundles on the moduli stack of stable curves332

3．The tangent bundle to moduli and related constructions344

4．The determinant of the cohomology and some applications347

5．The Deligne pairing366

6．The Picard group of moduli space379

7．Mumford's formula382

8．The Picard group of the hyperelliptic locus387

9．Bibliographical notes and further reading396

Chapter ⅩⅣ．Projectivity of the moduli space of stable curves399

1．Introduction399

2．A little invariant theory400

3．The invariant-theoretic stability of linearly stable smooth curves406

4．Numerical inequalities for families of stable curves414

5．Projectivity of moduli spaces425

6．Bibliographical notes and further reading437

Chapter ⅩⅤ．The Teichmüller point of view441

1．Introduction441

2．Teichmüller space and the mapping class group445

3．A little surface topology453

4．Quadratic differentials and Teichmüller deformations461

5．The geometry associated to a quadratic differential472

6．The proof of Teichmüller's uniqueness theorem479

7．Simple connectedness of the moduli stack of stable curves483

8．Going to the boundary of Teichmüller space485

9．Bibliographical notes and further reading497

10．Exercises498

Chapter ⅩⅥ．Smooth Galois covers of moduli spaces501

1．Introduction501

2．Level structures on smooth curves508

3．Automorphisms of stable curves515

4．Compactifying moduli of curves with level structure;a first attempt518

5．Admissible G-covers525

6．Automorphisms of admissible covers536

7．Smooth covers of ？g544

8．Totally unimodular lattices551

9．Smooth covers of ？g,n556

10．Bibliographical notes and further reading562

11．Exercises562

Chapter ⅩⅦ．Cycles in the moduli spaces of stable curves565

1．Introduction565

2．Algebraic cycles on quotients by finite groups566

3．Tautological classes on moduli spaces of curves570

4．Tautological relations and the tautological ring573

5．Mumford's relations for the Hodge classes585

6．Further considerations on cycles on moduli spaces596

7．The Chow ring of ？0,P599

8．Bibliographical notes and further reading604

9．Exercises605

Chapter ⅩⅧ．Cellular decomposition of moduli spaces609

1．Introduction609

2．The arc system complex613

3．Ribbon graphs616

4．The idea behind the cellular decomposition of Mg,n623

5．Uniformization624

6．Hyperbolic geometry627

7．The hyperbolic spine and the definition of ？636

8．The equivariant cellular decomposition of Teichmüller space643

9．Stable ribbon graphs648

10．Extending the cellular decomposition to a partial compactification of Teichmüller space652

11．The continuity of？655

12．Odds and ends661

13．Bibliographical notes and further reading665

Chapter ⅩⅨ．First consequences of the cellular decomposition667

1．Introduction667

2．The vanishing theorems for the rational homology of Mg,P670

3．Comparing the cohomology of ？g,n to the one of its boundary strata673

4．The second rational cohomology group of ？g,n676

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 spaces690

7．Combinatorial expression for the classes ψi694

8．A volume computation699

9．Bibliographical notes and further reading708

10．Exercises709

Chapter ⅩⅩ．Intersection theory of tautological classes717

1．Introduction717

2．Witten's generating series721

3．Virasoro operators and the KdV hierarchy726

4．The combinatorial identity729

5．Feynman diagrams and matrix models734

6．Kontsevich's matrix model and the equation L2Z＝0745

7．A nonvanishing theorem750

8．A brief review of equivariant cohomology and the virtual Euler-Poincaré characteristic754

9．The virtual Euler-Poincaré characteristic of Mg,n759

10．A very quick tour of Gromov-Witten invariants766

11．Bibliographical notes and further reading771

12．Exercises773

Chapter ⅩⅪ．Brill-Noether theory on a moving curve779

1．Introduction779

2．The relative Picard variety781

3．Brill-Noether varieties on moving curves788

4．Looijenga's vanishing theorem796

5．The Zariski tangent spaces to the Brill-Noether varieties802

6．The μ1 homomorphism808

7．Lazarsfeld's proof of Petri's conjecture814

8．The normal bundle and Horikawa's theory819

9．Ramification835

10．Plane curves845

11．The Hurwitz scheme and its irreducibility854

12．Plane curves and g？'s863

13．Unirationality results872

14．Bibliographical notes and further reading879

15．Exercises885

Bibliography903

Index945