概型理论是代数几何的基础，在代数几何的经典领域不变理论和曲线模中有了较好的发展。将代数数论和代数几何有机的结合起来，实现了早期数论学者们的愿望。这种结合使得数论中的一些主要猜测得以证明。

本书旨在建立起经典代数几何基本教程和概型理论之间的桥梁。例子讲解详实，努力挖掘定义背后的深层次东西。练习加深读者对内容的理解。学习本书的起点低，了解交换代数和代数变量的基本知识即可。本书揭示了概型和其他几何观点，如流形理论的联系。了解这些观点对学习本书是相当有益的，虽然不是必要。

Ⅰ Basic Definitions

 Ⅰ.1 Affine Schemes

Ⅰ.1.1 Schemes as Sets

Ⅰ.1.2 Schemes as Topological Spaces

Ⅰ.1.3 An Interlude on Sheaf Theory References for the Theory of Sheaves

Ⅰ.1.4 Schemes as Schemes (Structure Sheaves)

 Ⅰ.2 Schemes in General

Ⅰ.2.1 Subschemes

Ⅰ.2.2 The Local Ring at a Point

Ⅰ.2.3 Morphisms

Ⅰ.2.4 The Gluing Construction Projective Space

 Ⅰ.3 Relative Schemes

Ⅰ.3.1 Fibered Products

Ⅰ.3.2 The Category of S-Schemes

Ⅰ.3.3 Global Spec

 Ⅰ.4 The Functor of Points

Ⅱ Examples

 Ⅱ.1 Reduced Schemes over Algebraically Closed Fields

Ⅱ.1.1 Affine Spaces

Ⅱ.1.2 Local Schemes

 Ⅱ.2 Reduced Schemes over Non-Algebraically Closed Fields

 Ⅱ.3 Nonreduced Schemes

Ⅱ.3.1 Double Points

Ⅱ.3.2 Multiple Points

Degree and Multiplicity

Ⅱ.3.3 Embedded Points

Primary Decomposition

Ⅱ.3.4 Flat Families of Schemes

Limits

Examples

Flatness

Ⅱ.3.5 Multiple Lines

 Ⅱ.4 Arithmetic Schemes

Ⅱ.4.1 Spec Z

Ⅱ.4.2 Spec of the Ring of Integers in a Number Field

Ⅱ.4.3 Affine Spaces over Spec Z

Ⅱ.4.4 A Conic over Spec Z

Ⅱ.4.5 Double Points in Al

Ⅲ Projective Schemes

 Ⅲ.1 Attributes of Morphisms

Ⅲ.1.1 Finiteness Conditions

Ⅲ.1.2 Properness and Separation

 Ⅲ.2 Proj of a Graded Ring

Ⅲ.2.1 The Construction of Proj S

Ⅲ.2.2 Closed Subschemes of Proj R

Ⅲ.2.3 Global Proj

Proj of a Sheaf of Graded 0x-Algebras

The Projectivization P(ε) of a Coherent Sheaf ε

Ⅲ.2.4 Tangent Spaces and Tangent Cones

Affine and Projective Tangent Spaces

Tangent Cones

Ⅲ.2.5 Morphisms to Projective Space

Ⅲ.2.6 Graded Modules and Sheaves

Ⅲ.2.7 Grassmannians

Ⅲ.2.8 Universal Hypersurfaces

 Ⅲ.3 Invariants of Projective Schemes

Ⅲ.3.1 Hilbert Functions and Hilbert Polynomials

Ⅲ.3.2 Flatness Il: Families of Projective Schemes

Ⅲ.3.3 Free Resolutions

Ⅲ.3.4 Examples

Points in the Plane

Examples: Double Lines in General and in p3

Ⅲ.3.5 BEzout's Theorem

Multiplicity of Intersections

Ⅲ.3.6 Hilbert Series

Ⅳ Classical Constructions

 Ⅳ.1 Flexes of Plane Curves

Ⅳ.I.1 Definitions

Ⅳ.1.2 Flexes on Singular Curves

Ⅳ.1.3 Curves with Multiple Components

 Ⅳ.2 Blow-ups

Ⅳ.2.1 Definitions and Constructions

An Example: Blowing up the Plane

Definition of Blow-ups in General

The Blowup as Proj

Blow-ups along Regular Subschemes

Ⅳ.2.2 Some Classic Blow-Ups

Ⅳ.2.3 Blow-ups along Nonreduced Schemes

Blowing Up a Double Point

Blowing Up Multiple Points

The j-Fhnction

Ⅳ.2.4 Blow-ups of Arithmetic Schemes

Ⅳ.2.5 Project: Quadric and Cubic Surfaces as Blow-ups

 Ⅳ.3 Fano schemes

Ⅳ.3.1 Definitions

Ⅳ.3.2 Lines on Quadrics

Lines on a Smooth Quadric over an Algebraically

Closed Field

Lines on a Quadric Cone

A Quadric Degenerating to Two Planes

More Examples

Ⅳ.3.3 Lines on Cubic Surfaces

 Ⅳ.4 Forms

Ⅴ Local Constructions

 Ⅴ.1 Images

Ⅴ.1.1 The Image of a Morphism of Schemes

Ⅴ.1.2 Universal Formulas

Ⅴ.1.3 Fitting Ideals and Fitting Images

Fitting Ideals

Fitting Images

 Ⅴ.2 Resultants

Ⅴ.2.1 Definition of the Resultant

Ⅴ.2.2 Sylvester's Determinant

 Ⅴ.3 Singular Schemes and Discriminants

Ⅴ.3.1 Definitions

Ⅴ.3.2 Discriminants

Ⅴ.3.3 Examples

 Ⅴ.4 Dual Curves

Ⅴ.4.1 Definitions

Ⅴ.4.2 Duals of Singular Curves

Ⅴ.4.3 Curves with Multiple Components

 Ⅴ.5 Double Point Loci

Ⅵ Schemes and Functors

 Ⅵ.1 The Functor of Points

Ⅵ.I.1 Open and Closed Subfunetors

Ⅵ.1.2 K-Rational Points

Ⅵ.1.3 Tangent Spaces to a Functor

Ⅵ.1.4 Group Schemes

 Ⅵ.2 Characterization of a Space by its Functor of Points

Ⅵ.2.1 Characterization of Schemes among Functors

Ⅵ.2.2 Parameter Spaces

The Hilbert Scheme

Examples of Hilbert Schemes

Variations on the Hilbert Scheme Construction

Ⅵ.2.3 Tangent Spaces to Schemes in Terms of Their Functors of Points

Tangent Spaces to Hilbert Schemes

Tangent Spaces to Fano Schemes

Ⅵ.2.4 Moduli Spaces

References

Index