代数曲线几何

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.Ci 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 Teichmuller point of view

Chapter ⅩⅥ. Smooth Galois covers of moduli spaces

Chapter ⅩⅦ. Cycles in the moduli spaces of stable curves

Chapter ⅩⅧ. Cellular decomposition of moduli spaces

Chapter ⅩⅨ. First consequences of the cellular decomposition

Chapter ⅩⅩ. Intersection theory of tautological classes

Chapter ⅩⅩⅠ. Brill-Noether theory on a moving curve

《代数曲线几何（第2卷 第1分册）》是英文版的代数曲线的书。代数几何是现代数学的一个重要分支学科。它的基本研究对象是在任意维数的（仿射或射影）空间中，由若干个代数方程的公共零点所构成的集合的几何特性。这样的集合通常叫做代数簇，而这些方程叫做这个代数簇的定义方程组。