Basic Definitions 7
Ⅰ.1 Affine Schemes 7
Ⅰ.1.1 Schemes as Sets 9
Ⅰ.1.2 Schemes as Topological Spaces 10
Ⅰ.1.3 An Interlude on Sheaf Theory 11
References for the Theory of Sheaves 18
Ⅰ.1.4 Schemes as Schemes(Structure Sheaves) 18
Ⅰ.2 Schemes in General 21
Ⅰ.2.1 Subschemes 23
Ⅰ.2.2 The Local Ring at a Point 26
Ⅰ.2.3 Morphisms 28
Ⅰ.2.4 The Gluing Construction 33
Projective Space 34
Ⅰ.3 Relative Schemes 35
Ⅰ.3.1 Fibered Products 35
Ⅰ.3.2 The Category of S-Schemes 39
Ⅰ.3.3 Global Spec 40
Ⅰ.4 The Functor of Points 42
Ⅱ Examples 47
Ⅱ.1 Reduced Schemes over Algebraically Closed Fields 47
Ⅱ.1.1 Affine Spaces 47
Ⅱ.1.2 Local Schemes 50
Ⅱ.2 Reduced Schemes over Non-Algebraically Closed Fields 53
Ⅱ.3 Nonreduced Schemes 57
Ⅱ.3.1 Double Points 58
Ⅱ.3.2 Multiple Points 62
Degree and Multiplicity 65
Ⅱ.3.3 Embedded Points 66
Primary Decomposition 67
Ⅱ.3.4 Flat Families of Schemes 70
Limits 71
Examples 72
Flatness 75
Ⅱ.3.5 Multiple Lines 80
Ⅱ.4 Arithmetic Schemes 81
Ⅱ.4.1 Spec Z 82
Ⅱ.4.2 Spec of the Ring of Integers in a Number Field 82
Ⅱ.4.3 Affine Spaces over Spec Z 84
Ⅱ.4.4 A Conic over Spec Z 86
Ⅱ.4.5 Double Points in A1 Z 88
Ⅲ Projective Schemes 91
Ⅲ.1 Attributes of Morphisms 92
Ⅲ.1.1 Finiteness Conditions 92
Ⅲ.1.2 Properness and Separation 93
Ⅲ.2 Proj of a Graded Ring 95
Ⅲ.2.1 The Construction of Proj S 95
Ⅲ.2.2 Closed Subschemes of Proj R 100
Ⅲ.2.3 Global Proj 101
Proj of a Sheaf of Graded ？X-Algebras 101
The Projectivization P(？)of a Coherent Sheaf ？ 103
Ⅲ.2.4 Tangent Spaces and Tangent Cones 104
Affine and Projective Tangent Spaces 104
Tangent Cones 106
Ⅲ.2.5 Morphisms to Projective Space 110
Ⅲ.2.6 Graded Modules and Sheaves 118
Ⅲ.2.7 Grassmannians 119
Ⅲ.2.8 Universal Hypersurfaces 122
Ⅲ.3 Invariants of Projective Schemes 124
Ⅲ.3.1 Hilbert Functions and Hilbert Polynomials 125
Ⅲ.3.2 Flatness Ⅱ:Families of Projective Schemes 125
Ⅲ.3.3 Free Resolutions 127
Ⅲ.3.4 Examples 130
Points in the Plane 130
Examples:Double Lines in General and in P3 K 136
Ⅲ.3.5 Bézout's Theorem 140
Multiplicity of Intersections 146
Ⅲ.3.6 Hilbert Series 149
Ⅳ Classical Constructions 151
Ⅳ.1 Flexes of Plane Curves 151
Ⅳ.1.1 Definitions 151
Ⅳ.1.2 Flexes on Singular Curves 155
Ⅳ.1.3 Curves with Multiple Components 156
Ⅳ.2 Blow-ups 162
Ⅳ.2.1 Definitions and Constructions 162
An Example: Blowing up the Plane 163
Definition of Blow-ups in General 164
The Blowup as Proj 170
Blow-ups along Regular Subschemes 171
Ⅳ.2.2 Some Classic Blow-Ups 173
Ⅳ.2.3 Blow-ups along Nonreduced Schemes 179
Blowing Up a Double Point 179
Blowing Up Multiple Points 181
The j-Function 183
Ⅳ.2.4 Blow-ups of Arithmetic Schemes 184
Ⅳ.2.5 Project:Quadric and Cubic Surfaces as Blow-ups 190
Ⅳ.3 Fano schemes 192
Ⅳ.3.1 Definitions 192
Ⅳ.3.2 Lines on Quadrics 194
Lines on a Smooth Quadric over an Algebraically Closed Field 194
Lines on a Quadric Cone 196
A Quadric Degenerating to Two Planes 198
More Examples 201
Ⅳ.3.3 Lines on Cubic Surfaces 201
Ⅳ.4 Forms 204
Ⅴ Local Constructions 209
Ⅴ.1 Images 209
Ⅴ.1.1 The Image of a Morphism of Schemes 209
Ⅴ.1.2 Universal Formulas 213
Ⅴ.1.3 Fitting Ideals and Fitting Images 219
Fitting Ideals 219
Fitting Images 221
Ⅴ.2 Resultants 222
Ⅴ.2.1 Definition of the Resultant 222
Ⅴ.2.2 Sylvester's Determinant 224
Ⅴ.3 Singular Schemes and Discriminants 230
Ⅴ.3.1 Definitions 230
Ⅴ.3.2 Discriminants 232
Ⅴ.3.3 Examples 234
Ⅴ.4 Dual Curves 240
Ⅴ.4.1 Definitions 240
Ⅴ.4.2 Duals of Singular Curves 242
Ⅴ.4.3 Curves with Multiple Components 242
Ⅴ.5 Double Point Loci 246
Ⅵ Schemes and Functors 251
Ⅵ.1 The Functor of Points 252
Ⅵ.1.1 Open and Closed Subfunctors 254
Ⅵ.1.2 K-Rational Points 256
Ⅵ.1.3 Tangent Spaces to a Functor 256
Ⅵ.1.4 Group Schemes 258
Ⅵ.2 Characterization of a Space by its Functor of Points 259
Ⅵ.2.1 Characterization of Schemes among Functors 259
Ⅵ.2.2 Parameter Spaces 262
The Hilbert Scheme 262
Examples of Hilbert Schemes 264
Variations on the Hilbert Scheme Construction 265
Ⅵ.2.3 Tangent Spaces to Schemes in Terms of Their Func-tors of Points 267
Tangent Spaces to Hilbert Schemes 267
Tangent Spaces to Fano Schemes 271
Ⅵ.2.4 Moduli Spaces 274
References 279
Index 285