概型理論是代數幾何的基礎，在代數幾何的經典領域不變理論和曲線模中有了較好的發展。將代數數論和代數幾何有機的結合起來，實現了早期數論學者們的願望。這種結合使得數論中的一些主要猜測得以證明。
本書旨在建立起經典代數幾何基本教程和概型理論之間的橋梁。例子講解詳實，努力挖掘定義背後的深層次東西。練習加深讀者對內容的理解。學習本書的起點低，了解交換代數和代數變量的基本知識即可。本書揭示了概型和其他幾何觀點，如流形理論的聯繫。了解這些觀點對學習本書是相當有益的，雖然不是必要。

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