Book 2: Schemes and Varieties
5 Schemes
1 The Spec of a Ring
1.1 Definition of Spec A
1.2 Properties of Points of SpecA
1.3 The Zariski Topology of Spec A
1.4 Irreducibility,Dimension
1.5 Exercises to Section 1
2 Sheaves
2.1 Presheaves
2.2 The Structure Presheaf
2.3 Sheaves
2.4 Stalks of a Sheaf
2.5 Exercises to Section 2
3 Schemes
3.1 Definition of a Scheme
3.2 Glueing Schemes
3.3 Closed Subschemes
3.4 Reduced Schemes and Nilpotents
3.5 Finiteness Conditions
3.6 Exercises to Section 3
4 Products of Schemes
4.1 Definition of Product
4.2 Group Schemes
4.3 Separatedness
4.4 Exercises to Section 4
6 Varieties
1 Definitions and Examples
1.1 Definitions
1.2 Vector Bundles
1.3 Vector Bundles and Sheaves
1.4 Divisors and Line Bundles
1.5 Exercises to Section 1
2 Abstract and Quasiprojective Varieties
2.1 Chow's Lemma
2.2 Blowup Along a Subvariety
2.3 Example of Non-quasiprojective Variety
2.4 Criterions for Projectivity
2.5 Exercises to Section 2
3 Coherent Sheaves
3.1 Sheaves of Ox-Modules
3.2 Coherent Sheaves
3.3 Devissage of Coherent Sheaves
3.4 The Finiteness Theorem
3.5 Exercises to Section 3
4 Classification of Geometric Objects and Universal Schemes
4.1 Schemes and Functors
4.2 The Hilbert Polynomial
4.3 Flat Families
4.4 The Hilbert Scheme
