概型的几何（英文版）
目 录内容简介
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
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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
查看完整
目 录内容简介
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。
《概型的几何(英文版)》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《概型的几何(英文版)》的起点低，了解交换代数和代数变量的基本知识即可。《概型的几何(英文版)》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《概型的几何(英文版)》是相当有益的，虽然不是必要。目次：基本定义；例子；射影概型；经典结构；局部结构；概型和函子。
《概型的几何(英文版)》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《概型的几何(英文版)》的起点低，了解交换代数和代数变量的基本知识即可。《概型的几何(英文版)》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《概型的几何(英文版)》是相当有益的，虽然不是必要。目次：基本定义；例子；射影概型；经典结构；局部结构；概型和函子。
目 录内容简介
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
1II.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 BEzouts 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-Function
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.I.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：l Definition of the Resultant
V.2.2 Sylvesters 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 Subfunctors
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 ~nctor 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 Func
tors of Points
Tangent Spaces to Hilbert Schemes
Tangent Spaces to Fano Schemes
VI.2.4 Moduli Spaces
References
Index
^ 收 起
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
1II.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 BEzouts 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-Function
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.I.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：l Definition of the Resultant
V.2.2 Sylvesters 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 Subfunctors
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 ~nctor 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 Func
tors of Points
Tangent Spaces to Hilbert Schemes
Tangent Spaces to Fano Schemes
VI.2.4 Moduli Spaces
References
Index
^ 收 起
目 录内容简介
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。
《概型的几何(英文版)》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《概型的几何(英文版)》的起点低，了解交换代数和代数变量的基本知识即可。《概型的几何(英文版)》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《概型的几何(英文版)》是相当有益的，虽然不是必要。目次：基本定义；例子；射影概型；经典结构；局部结构；概型和函子。
《概型的几何(英文版)》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《概型的几何(英文版)》的起点低，了解交换代数和代数变量的基本知识即可。《概型的几何(英文版)》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《概型的几何(英文版)》是相当有益的，虽然不是必要。目次：基本定义；例子；射影概型；经典结构；局部结构；概型和函子。
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