概型的幾何

《 概型的幾何 》是 第1版 (2010年1月1日) 世界圖書出版公司; 出版的圖書

基本介紹

• 書名：概型的幾何
• 頁數：294頁
• 出版社：世界圖書出版公司;
• 出版時間： 第1版 (2010年1月1日)

圖書信息

外文書名: The Geometry of Schemes

正文語種: 英語

開本: 24

ISBN: 7510004748, 9787510004742

條形碼: 9787510004742

尺寸: 22.2 x 14.8 x 1.4 cm

重量: 399 g

內容簡介

《概型的幾何(英文版)》內容簡介：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。

《概型的幾何(英文版)》旨在建立起經典代數幾何基本教程和概型理論之間的橋樑。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《概型的幾何(英文版)》的起點低，了解交換代數和代數變數的基本知識即可。《概型的幾何(英文版)》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《概型的幾何(英文版)》是相當有益的，雖然不是必要。目次：基本定義；例子；射影概型；經典結構；局部結構；概型和函子。

目錄

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 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-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 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 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