Schemes - D. Eisenbud, J. Harris
Basic Definitions 7
1.1 Ane Schemes ......................... 7
1.1.1 Schemes as Sets .................... 9
1.1.2 Schemes as Topological Spaces ............ 10
1.1.3 An Interlude on Sheaf Theory ............ 11
References for the Theory of Sheaves ........ 18
1.1.4 Schemes as Schemes (Structure Sheaves) ...... 18
1.2 Schemes in General ...................... 21
1.2.1 Subschemes ...................... 23
1.2.2 The Local Ring at a Point .............. 26
1.2.3 Morphisms ....................... 28
1.2.4 The Gluing Construction ............... 33
Projective Space .................... 34
1.3 Relative Schemes ........................ 35
1.3.1 Fibered Products ................... 35
1.3.2 The Category of S-Schemes ............. 39
1.3.3 Global Spec ...................... 40
1.4 The Functor of Points ..................... 42
Examples 47
II.1 Reduced Schemes over Algebraically Closed Fields ..... 47
II.l.1 Afflne Spaces ...................... 47
II.1.2 Local Schemes ..................... 50
II.2 Reduced Schemes over Non-Algebraically Closed Fields . . 53
viii Contents
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 SpecZ .................. 86
II.4.5 Double Points in A .................. 88
III Projective Schemes 91
III.1 Attributes of Morphisms ................... 92
III.l.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 $ .............. 95
III.2.2 Closed Subschemes of Proj R ............. 100
III.2.3 Global Proj ...................... 101
Proj of a Sheaf of Graded fix-Algebras ....... 101
The Projectivization ?() 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 ? . . . 136
III.3.5 Bzout's Theorem ................... 140
Multiplicity of Intersections .............. 146
III.3.6 Hilbert Series ..................... 149
Contents ix
IV Classical Constructions 151
IV.1 Flexes of Plane Curves .................... 151
IV.I.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 .................. 169
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
x Contents
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
VIA The Functor of Points ..................... 252
VI.I.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
Contents
Basic Definitions 7
1.1 Ane Schemes ......................... 7
1.1.1 Schemes as Sets .................... 9
1.1.2 Schemes as Topological Spaces ............ 10
1.1.3 An Interlude on Sheaf Theory ............ 11
References for the Theory of Sheaves ........ 18
1.1.4 Schemes as Schemes (Structure Sheaves) ...... 18
1.2 Schemes in General ...................... 21
1.2.1 Subschemes ...................... 23
1.2.2 The Local Ring at a Point .............. 26
1.2.3 Morphisms ....................... 28
1.2.4 The Gluing Construction ............... 33
Projective Space .................... 34
1.3 Relative Schemes ........................ 35
1.3.1 Fibered Products ................... 35
1.3.2 The Category of S-Schemes ............. 39
1.3.3 Global Spec ...................... 40
1.4 The Functor of Points ..................... 42
Examples 47
II.1 Reduced Schemes over Algebraically Closed Fields ..... 47
II.l.1 Afflne Spaces ...................... 47
II.1.2 Local Schemes ..................... 50
II.2 Reduced Schemes over Non-Algebraically Closed Fields . . 53
viii Contents
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 SpecZ .................. 86
II.4.5 Double Points in A .................. 88
III Projective Schemes 91
III.1 Attributes of Morphisms ................... 92
III.l.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 $ .............. 95
III.2.2 Closed Subschemes of Proj R ............. 100
III.2.3 Global Proj ...................... 101
Proj of a Sheaf of Graded fix-Algebras ....... 101
The Projectivization ?() 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 ? . . . 136
III.3.5 Bzout's Theorem ................... 140
Multiplicity of Intersections .............. 146
III.3.6 Hilbert Series ..................... 149
Contents ix
IV Classical Constructions 151
IV.1 Flexes of Plane Curves .................... 151
IV.I.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 .................. 169
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
x Contents
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
VIA The Functor of Points ..................... 252
VI.I.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
