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
4.5 Exercises to Section 4 Book 3: Complex Algebraic Varieties and Complex Manifolds 7 The Topology of Algebraic Varieties 1 The Complex Topology 1.1 Definitions 1.2 Algebraic Varieties as Differentiable Manifolds; Orientation 1.3 Homology of Nonsingular Projective Varieties 1.4 Exercises to Section 1 2 Connectedness 2.1 Preliminary Lemmas 2.2 The First Proof of the Main Theorem 2.3 The Second Proof 2.4 Analytic Lemmas 2.5 Connectedness of Fibres 2.6 Exercises to Section 2 3 The Topology of Algebraic Curves 3.1 Local Structure of Morphisms 3.2 Triangulation of Curves 3.3 Topological Classification of Curves 3.4 Combinatorial Classification of Surfaces 3.5 The Topology of Singularities of Plane Curves 3.6 Exercises to Section 3 …… Book 1: Varieties in Projective space Algebraic Appendix References Index
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