The Geometry of Schemes

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.l 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 A1
III Projective Schemes
III.l 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 O_x-Algebras
The Projectivization P(E) of a Coherent Sheaf E
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 II: Families of Projective Schemes
III.3.3 Free Resolutions
III.3.4 Examples
Points in the Plane
Examples: Double Lines in General and in IP'k
III.3.5 Bezout's Theorem
Multiplicity of Intersections
III.3.6 Hilbert Series
IV Classical Constructions
IV.1 Flexes of Plane Curves
IV.1.1 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.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.l The Functor of Points
VI.l.l 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 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