This volume is devoted to the foundations of the theory of moduli of algebraic curves defined over the complex numbers. The first volume was almost exclusively concerned with the geometry on a fixed, smooth curve. At the time it was published, the local deformation theory of a smooth curve was well understood, but the study of the geometry of global moduli was in its early stages. This study has since undergone explosive development and continues to do so. There are two reasons for this; one predictable at the time of the first volume, the other not.
The predictable one was the intrinsic algebro-geometric interest in the moduli of curves; this has certainly turned out to be the case. The other is the external influence from physics. Because of this confluence, the subject has developed in ways that are incredibly richer than could have been imagined at the time of writing of Volume I.
When this volume, GAC II, was planned it was envisioned that the cen- terpiece would be the study of linear series on a general or variable curve, culminating in a proof of the Petri conjecture. This is still an important part of the present volume, but it is not the central aspect. Rather, the main purpose of the book is to provide comprehensive and detailed foundations for the theory of the moduli of algebraic curves. In addition, we feel that a very important, perhaps distinguishing, aspect of GAC II is the blending of the multiple perspectives-algebro-geometric, complex-analytic, topological, and combinatorial-that are used for the study of the moduli of curves.

Guide for the Reader
List of Symbols

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 analvtic SDace
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.he 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 aDDlications
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
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 Teichmuller point of view
1.Introduction
2.Teichmuller space and the mapping class group
3.A little surface topology
4.Quadratic differentials and Teichmuller deformations
5.The geometry associated to a quadratic differential
6.The proof of Teichmuller's uniqueness theorem
7.Simple connectedness of the moduli stack of stable curves
8.Going to the boundary of Teichmuller 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 Mq
8.Totally unimodular lattices
9.Smooth covers of Mg，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 MO，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 ofψ
8.The equivariant cellular decomposition of Teichmuller space
9.Stable ribbon graphs
10.Extending the cellular decomposition to a partial compactification of Teichmuller space
11.The continuity of ψ
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 Mg，n to the one of its boundary strata
4.The second rational cohomology group of Mg，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 eauation L2Z=0
7.A nonvanishing theorem
8.A brief review of equivariant cohomology and the virtual Euler-Poincare characteristic
9.The virtual Euler-Poincare 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 g1d's
13.Unirationality results
14.Bibliographical notes and further reading
15.Exercises
Bibliography
Index