概型的幾何=The Geometry of Schemes by?

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