Chapter Ⅸ．The Hilbert Scheme

1．Introduction

2．The idea of the Hilbert scheme

3．Flatness

4．Construction of the Hilbert scheme

5．The characteristic system

6．Mumford's example

7．Variants of the Hilbert scheme

8．Tangent space computations

9．Cn families of projective manifolds

10．Bibliographical notes and further reading

11．Exercises

Chapter Ⅹ．Nodal curves

1．Introduction

2．Elementary theory of nodal curves

3．Stable curves

4．Stable reduction

5．Isomorphisms of families of stable curves

6．The stable model,contraction,and projection

7．Clutching

8．Stabilization

9．Vanishing cycles and the Picard-Lefschetz transformation

10．Bibliographical notes and further reading

11．Exercises

Chapter Ⅺ．Elementary deformation theory and some applications

1．Introduction

2．Deformations of manifolds

3．Deformations of nodal curves

4．The concept of Kuranishi family

5．The Hilbert scheme of v-canonical curves

6．Construction of Kuranishi families

7．The Kuranishi family and continuous deformations

8．The period map and the local Torelli theorem

9．Curvature of the Hodge bundles

10．Deformations of symmetric products

11．Bibliographical notes and further reading

Chapter Ⅻ．The moduli space of stable curves

1．Introduction

2．Construction of moduli space as an analytic space

3．Moduli spaces as algebraic spaces

4．The moduli space of curves as an orbifold

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

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

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

8．Deligne-Mumford stacks

9．Back to algebraic spaces

10．The universal curve,projections and clutchings

11．Bibliographical notes and further reading

12．Exercises

Chapter ⅩⅢ．Line bundles on moduli

1．Introduction

2．Line bundles on the moduli stack of stable curves

3．The tangent bundle to moduli and related constructions

4．The determinant of the cohomology and some applications

5．The Deligne pairing

6．The Picard group of moduli space

7．Mumford's formula

8．The Picard group of the hyperelliptic locus

9．Bibliographical notes and further reading

Chapter ⅩⅣ．Projectivity of the moduli space of stable curves

1．Introduction

2．A little invariant theory

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

4．Numerical inequalities for families of stable curves

5．Projectivity of moduli spaces

6．Bibliographical notes and further reading

Chapter ⅩⅤ．The Teichmüller point of view

1．Introduction

2．Teichmüller space and the mapping class group

3．A little surface topology

4．Quadratic differentials and Teichmüller deformations

5．The geometry associated to a quadratic differential

6．The proof of Teichmüller's uniqueness theorem

7．Simple connectedness of the moduli stack of stable curves

8．Going to the boundary of Teichmüller space

9．Bibliographical notes and further reading

10．Exercises

Chapter ⅩⅥ．Smooth Galois covers of moduli spaces

1．Introduction

2．Level structures on smooth curves

3．Automorphisms of stable curves

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

5．Admissible G-covers

6．Automorphisms of admissible covers

7．Smooth covers of ？g

8．Totally unimodular lattices

9．Smooth covers of ？g,n

10．Bibliographical notes and further reading

11．Exercises

Chapter ⅩⅦ．Cycles in the moduli spaces of stable curves

1．Introduction

2．Algebraic cycles on quotients by finite groups

3．Tautological classes on moduli spaces of curves

4．Tautological relations and the tautological ring

5．Mumford's relations for the Hodge classes

6．Further considerations on cycles on moduli spaces

7．The Chow ring of ？0,P

8．Bibliographical notes and further reading

9．Exercises

Chapter ⅩⅧ．Cellular decomposition of moduli spaces

1．Introduction

2．The arc system complex

3．Ribbon graphs

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

5．Uniformization

6．Hyperbolic geometry

7．The hyperbolic spine and the definition of ？

8．The equivariant cellular decomposition of Teichmüller space

9．Stable ribbon graphs

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

11．The continuity of？

12．Odds and ends

13．Bibliographical notes and further reading

Chapter ⅩⅨ．First consequences of the cellular decomposition

1．Introduction

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

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

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

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

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

7．Combinatorial expression for the classes ψi

8．A volume computation

9．Bibliographical notes and further reading

10．Exercises

Chapter ⅩⅩ．Intersection theory of tautological classes

1．Introduction

2．Witten's generating series

3．Virasoro operators and the KdV hierarchy

4．The combinatorial identity

5．Feynman diagrams and matrix models

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

7．A nonvanishing theorem

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

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

10．A very quick tour of Gromov-Witten invariants

11．Bibliographical notes and further reading

12．Exercises

Chapter ⅩⅪ．Brill-Noether theory on a moving curve

1．Introduction

2．The relative Picard variety

3．Brill-Noether varieties on moving curves

4．Looijenga's vanishing theorem

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

6．The μ1 homomorphism

7．Lazarsfeld's proof of Petri's conjecture

8．The normal bundle and Horikawa's theory

9．Ramification

10．Plane curves

11．The Hurwitz scheme and its irreducibility

12．Plane curves and g？'s

13．Unirationality results

14．Bibliographical notes and further reading

15．Exercises

Bibliography

Index