概型的几何

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

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。