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Basic Definitions7

Ⅰ.1 Affine Schemes7

Ⅰ.1.1 Schemes as Sets9

Ⅰ.1.2 Schemes as Topological Spaces10

Ⅰ.1.3 An Interlude on Sheaf Theory11

References for the Theory of Sheaves18

Ⅰ.1.4 Schemes as Schemes(Structure Sheaves)18

Ⅰ.2 Schemes in General21

Ⅰ.2.1 Subschemes23

Ⅰ.2.2 The Local Ring at a Point26

Ⅰ.2.3 Morphisms28

Ⅰ.2.4 The Gluing Construction33

Projective Space34

Ⅰ.3 Relative Schemes35

Ⅰ.3.1 Fibered Products35

Ⅰ.3.2 The Category of S-Schemes39

Ⅰ.3.3 Global Spec40

Ⅰ.4 The Functor of Points42

Ⅱ Examples47

Ⅱ.1 Reduced Schemes over Algebraically Closed Fields47

Ⅱ.1.1 Affine Spaces47

Ⅱ.1.2 Local Schemes50

Ⅱ.2 Reduced Schemes over Non-Algebraically Closed Fields53

Ⅱ.3 Nonreduced Schemes57

Ⅱ.3.1 Double Points58

Ⅱ.3.2 Multiple Points62

Degree and Multiplicity65

Ⅱ.3.3 Embedded Points66

Primary Decomposition67

Ⅱ.3.4 Flat Families of Schemes70

Limits71

Examples72

Flatness75

Ⅱ.3.5 Multiple Lines80

Ⅱ.4 Arithmetic Schemes81

Ⅱ.4.1 Spec Z82

Ⅱ.4.2 Spec of the Ring of Integers in a Number Field82

Ⅱ.4.3 Affine Spaces over Spec Z84

Ⅱ.4.4 A Conic over Spec Z86

Ⅱ.4.5 Double Points in A1 Z88

Ⅲ Projective Schemes91

Ⅲ.1 Attributes of Morphisms92

Ⅲ.1.1 Finiteness Conditions92

Ⅲ.1.2 Properness and Separation93

Ⅲ.2 Proj of a Graded Ring95

Ⅲ.2.1 The Construction of Proj S95

Ⅲ.2.2 Closed Subschemes of Proj R100

Ⅲ.2.3 Global Proj101

Proj of a Sheaf of Graded ？X-Algebras101

The Projectivization P(？)of a Coherent Sheaf ？103

Ⅲ.2.4 Tangent Spaces and Tangent Cones104

Affine and Projective Tangent Spaces104

Tangent Cones106

Ⅲ.2.5 Morphisms to Projective Space110

Ⅲ.2.6 Graded Modules and Sheaves118

Ⅲ.2.7 Grassmannians119

Ⅲ.2.8 Universal Hypersurfaces122

Ⅲ.3 Invariants of Projective Schemes124

Ⅲ.3.1 Hilbert Functions and Hilbert Polynomials125

Ⅲ.3.2 Flatness Ⅱ:Families of Projective Schemes125

Ⅲ.3.3 Free Resolutions127

Ⅲ.3.4 Examples130

Points in the Plane130

Examples:Double Lines in General and in P3 K136

Ⅲ.3.5 Bézout's Theorem140

Multiplicity of Intersections146

Ⅲ.3.6 Hilbert Series149

Ⅳ Classical Constructions151

Ⅳ.1 Flexes of Plane Curves151

Ⅳ.1.1 Definitions151

Ⅳ.1.2 Flexes on Singular Curves155

Ⅳ.1.3 Curves with Multiple Components156

Ⅳ.2 Blow-ups162

Ⅳ.2.1 Definitions and Constructions162

An Example: Blowing up the Plane163

Definition of Blow-ups in General164

The Blowup as Proj170

Blow-ups along Regular Subschemes171

Ⅳ.2.2 Some Classic Blow-Ups173

Ⅳ.2.3 Blow-ups along Nonreduced Schemes179

Blowing Up a Double Point179

Blowing Up Multiple Points181

The j-Function183

Ⅳ.2.4 Blow-ups of Arithmetic Schemes184

Ⅳ.2.5 Project:Quadric and Cubic Surfaces as Blow-ups190

Ⅳ.3 Fano schemes192

Ⅳ.3.1 Definitions192

Ⅳ.3.2 Lines on Quadrics194

Lines on a Smooth Quadric over an Algebraically Closed Field194

Lines on a Quadric Cone196

A Quadric Degenerating to Two Planes198

More Examples201

Ⅳ.3.3 Lines on Cubic Surfaces201

Ⅳ.4 Forms204

Ⅴ Local Constructions209

Ⅴ.1 Images209

Ⅴ.1.1 The Image of a Morphism of Schemes209

Ⅴ.1.2 Universal Formulas213

Ⅴ.1.3 Fitting Ideals and Fitting Images219

Fitting Ideals219

Fitting Images221

Ⅴ.2 Resultants222

Ⅴ.2.1 Definition of the Resultant222

Ⅴ.2.2 Sylvester's Determinant224

Ⅴ.3 Singular Schemes and Discriminants230

Ⅴ.3.1 Definitions230

Ⅴ.3.2 Discriminants232

Ⅴ.3.3 Examples234

Ⅴ.4 Dual Curves240

Ⅴ.4.1 Definitions240

Ⅴ.4.2 Duals of Singular Curves242

Ⅴ.4.3 Curves with Multiple Components242

Ⅴ.5 Double Point Loci246

Ⅵ Schemes and Functors251

Ⅵ.1 The Functor of Points252

Ⅵ.1.1 Open and Closed Subfunctors254

Ⅵ.1.2 K-Rational Points256

Ⅵ.1.3 Tangent Spaces to a Functor256

Ⅵ.1.4 Group Schemes258

Ⅵ.2 Characterization of a Space by its Functor of Points259

Ⅵ.2.1 Characterization of Schemes among Functors259

Ⅵ.2.2 Parameter Spaces262

The Hilbert Scheme262

Examples of Hilbert Schemes264

Variations on the Hilbert Scheme Construction265

Ⅵ.2.3 Tangent Spaces to Schemes in Terms of Their Func-tors of Points267

Tangent Spaces to Hilbert Schemes267

Tangent Spaces to Fano Schemes271

Ⅵ.2.4 Moduli Spaces274

References279

Index285