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Guide for the ReaderList of SymbolsChapter Ⅸ.The Hilbert Scheme1.Introduction2.The idea of the Hilbert scheme3.Flatness4.Construction of the Hilbert scheme5.The characteristic system6.Mumford's example7.Variants of the Hilbert scheme8.Tangent space computations9.Cn families of projective manifolds10.Bibliographical notes and further reading11.ExercisesChapter Ⅹ.Nodal curves1.Introduction2.Elementary theory of nodal curves3.Stable curves4.Stable reduction5.Isomorphisms of families of stable curves6.The stable model， contraction， and projection7.Clutching8.Stabilization9.Vanishing cycles and the Picard-Lefschetz transformation10.Bibliographical notes and further reading11.ExercisesChapter Ⅺ.Elementary deformation theory and some applications1.Introduction2.Deformations of manifolds3.Deformations of nodal curves4.The concept of Kuranishi family.5.The Hilbert scheme of v-canonical curves6.Construction of Kuranishi families7.The Kuranishi family and continuous deformations8.The period map and the local Torelli theorem9.Curvature of the Hodge bundles10.Deformations of symmetric products11.Bibliographical notes and further readingChapter ⅩⅡ.The moduli space of stable curves1.Introduction2.Construction of' moduli space as an analvtic SDace3.Moduli spaces as algebraic spaces4.The moduli space of curves as an orbifold5.The moduli space of curves as a stack， I.6.he classical theory of descent for quasi-coherent sheaves7.The moduli space of curves as a stack， Ⅱ8.Deligne-Mumford stacks9.Back to algebraic spaces10.The universal curve， projections and clutchings11.Bibliographical notes and further reading12.ExercisesChapter ⅩⅢ.Line bundles on moduli1.Introduction2.Line bundles on the moduli stack of stable curves3.The tangent bundle to moduli and related constructions4.The determinant of the cohomology and some aDDlications5.The Deligne pairing6.The Picard group of moduli space7.Mumford's formula8.The Picard group of the hyperelliptic locus9.Bibliographical notes and further readingChapter ⅩⅣ.Projectivity of the moduli space of stable1.Introduction2.A little invariant theory3.The invariant-theoretic stability of linearly stable smooth curves4.Numerical inequalities for families of stable curves5.Projectivity of moduli spaces6.Bibliographical notes and further readingChapter ⅩⅤ.The Teichmuller point of view1.Introduction2.Teichmuller space and the mapping class group3.A little surface topology4.Quadratic differentials and Teichmuller deformations5.The geometry associated to a quadratic differential6.The proof of Teichmuller's uniqueness theorem7.Simple connectedness of the moduli stack of stable curves8.Going to the boundary of Teichmuller space9.Bibliographical notes and further reading10.ExercisesChapter ⅩⅥ.Smooth Galois covers of moduli spaces1.Introduction2.Level structures on smooth curves3.Automorphisms of stable curves4.Compactifying moduli of curves with level structure， a first attempt5.Admissible G-covers6.Automorphisms of admissible covers7.Smooth covers of Mq8.Totally unimodular lattices9.Smooth covers of Mg，n10.Bibliographical notes and further reading11.ExercisesChapter ⅩⅦ.Cycles in the moduli spaces of stable curves1.Introduction2.Algebraic cycles on quotients by finite groups3.Tautological classes on moduli spaces of curves4.Tautological relations and the tautological ring5.Mumford's relations for the Hodge classes6.Further considerations on cycles on moduli spaces7.The Chow ring of MO，P8.Bibliographical notes and further reading9.ExercisesChapter ⅩⅧ.Cellular decomposition of moduli spaces1.Introduction2.The arc system complex3.Ribbon graphs4.The idea behind the cellular decomposition of Mg，n5.Uniformization6.Hyperbolic geometry7.The hyperbolic spine and the definition ofψ8.The equivariant cellular decomposition of Teichmuller space9.Stable ribbon graphs10.Extending the cellular decomposition to a partial compactification of Teichmuller space11.The continuity of ψ12.Odds and ends13.Bibliographical notes and further readingChapter ⅪⅩ.First consequences of the cellular decomposition1.Introduction2.The vanishing theorems for the rational homology of Mg，p3.Comparing the cohomology of Mg，n to the one of its boundary strata4.The second rational cohomology group of Mg，n5.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 spaces7.Combinatorial expression for the classes ψi8.A volume computation9.Bibliographical notes and further reading10.ExercisesChapter ⅩⅩ.Intersection theory of tautological classes1.Introduction2.Witten's generating series3.Virasoro operators and the KdV hierarchy4.The combinatorial identity5.Feynman diagrams and matrix models6.Kontsevich's matrix model and the eauation L2Z=07.A nonvanishing theorem8.A brief review of equivariant cohomology and the virtual Euler-Poincare characteristic9.The virtual Euler-Poincare characteristic of Mg，n10.A very quick tour of Gromov-Witten invariants11.Bibliographical notes and further reading12.ExercisesChapter ⅩⅪ.Brill-Noether theory on a moving curve1.Introduction2.The relative Picard variety3.Brill-Noether varieties on moving curves4.Looijenga's vanishing theorem5.The Zariski tangent spaces to the Brill-Noether varieties6.The μ1 homomorphism7.Lazarsfeld's proof of Petri's conjecture8.The normal bundle and Horikawa's theory9. Ramification10.Plane curves11.The Hurwitz scheme and its irreducibility12.Plane curves and g1d's13.Unirationality results14.Bibliographical notes and further reading15.ExercisesBibliographyIndex

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.