Book 1: Varieties in Projective Space 1 Basic Notions 1 Algebraic Curves in the Plane 1.1 Plane Curves 1.2 Rational Curves 1.3 Relation with Field Theory 1.4 Rational Maps 1.5 Singular and Nonsingular Points 1.6 The Projective Plane 1.7 Exercises to Section 1 2 Closed Subsets of Affine Space 2.1 Definition of Closed Subsets 2.2 Regular Functions on a Closed Subset 2.3 Regular Maps 2.4 Exercises to Section 2 3 Rational Functions 3.1 Irreducible Algebraic Subsets 3.2 Rational Functions 3.3 Rational Maps 3.4 Exercises to Section 3 4 Quasiprojective Varieties 4.1 Closed Subsets of Projective Space 4.2 Regular Functions 4.3 Rational Functions 4.4 Examples of Regular Maps 4.5 Exercises to Section 4 5 Products and Maps of Quasiprojective Varieties 5.1 Products 5.2 The Image of a Projective Variety is Closed 5.3 Finite Maps 5.4 Noether Normalisation 5.5 Exercises to Section 5 6 Dimension 6.1 Definition of Dimension 6.2 Dimension of Intersection with a Hypersurface 6.3 The Theorem on the Dimension of Fibres 6.4 Lines on Surfaces 6.5 Exercises to Section 6 2 Local Properties 1 Singular and Nonsingular Points 1.1 The Local Ring of a Point 1.2 The Tangent Space 1.3 Intrinsic Nature of the Tangent Space 1.4 Singular Points 1.5 The Tangent Cone 1.6 Exercises to Section 1 2 Power Series Expansions 2.1 Local Parameters at a Point 2.2 Power Series Expansions 2.3 Varieties over the Reals and the Complexes
2.4 Exercises to Section 2 3 Properties of Nonsingular Points 3.1 Codimension 1 Subvarieties 3.2 Nonsingular Subvarieties 3.3 Exercises to Section 3 4 The Structure of Birational Maps 4.1 Blowup in Projective Space 4.2 Local Blowup 4.3 Behaviour of a Subvariety Under a Blowup 4.4 Exceptional Subvarieties 4.5 Isomorphism and Birational Equivalence 4.6 Exercises to Section 4 5 Normal Varieties 5.1 Normal Varieties 5.2 Normalisation of an Affine Variety 5.3 Normalisation of a Curve 5.4 Projective Embedding of Nonsingular Varieties 5.5 Exercises to Section 5 6 Singularities of a Map 6.1 Irreducibility 6.2 Nonsingularity 6.3 Ramification 6.4 Examples 6.5 Exercises to Section 6 …… Book 2: Schemes and Varieties Book 3: Complex Algebraic Varieties and Complex Manifolds
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