Basic Algebraic Geometry 1

I. 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.- Exercises to §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.- Exercises to §2.- 3. Rational Functions.- 3.1. Irreducible Algebraic Subsets.- 3.2. Rational Functions.- 3.3. Rational Maps.- Exercises to §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.- Exercises to §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.- Exercises to §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.- Exercises to §6.- II. 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.- Exercises to §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.- Exercises to §2.- 3. Properties of Nonsingular Points.- 3.1. Codimension 1 Subvarieties.- 3.2. Nonsingular Subvarieties.- Exercises to §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.- Exercises to §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.- Exercises to §5.- 6. Singularities of a Map.- 6.1. Irreducibility.- 6.2. Nonsingularity.- 6.3. Ramification.- 6.4. Examples.- Exercises to §6.- III. Divisors and Differential Forms.- 1. Divisors.- 1.1. The Divisor of a Function.- 1.2. Locally Principal Divisors.- 1.3. Moving the Support of a Divisor away from a Point ....- 1.4. Divisors and Rational Maps.- 1.5. The Linear System of a Divisor.- 1.6. Pencil of Conics over ?1.- Exercises to §1.- 2. Divisors on Curves.- 2.1. The Degree of a Divisor on a Curve.- 2.2. Bézout's Theorem on a Curve.- 2.3. The Dimension of a Divisor.- Exercises to §2.- 3. The Plane Cubic.- 3.1. The Class Group.- 3.2. The Group Law.- 3.3. Maps.- 3.4. Applications.- 3.5. Algebraically Nonclosed Field.- Exercises to §3.- 4. Algebraic Groups.- 4.1. Algebraic Groups.- 4.2. Quotient Groups and Chevalley's Theorem.- 4.3. Abelian Varieties.- 4.4. The Picard Variety.- Exercises to §4.- 5. Differential Forms.- 5.1. Regular Differential 1-forms.- 5.2. Algebraic Definition of the Module of Differentials.- 5.3. Differential p-forms.- 5.4. Rational Differential Forms.- Exercises to §5.- 6. Examples and Applications of Differential Forms.- 6.1. Behaviour Under Maps.- 6.2. Invariant Differential Forms on a Group.- 6.3. The Canonical Class.- 6.4. Hypersurfaces.- 6.5. Hyperelliptic Curves.- 6.6. The Riemann-Roch Theorem for Curves.- 6.7. Projective Embedding of a Surface.- Exercises to §6.- IV. Intersection Numbers.- 1. Definition and Basic Properties.- 1.1. Definition of Intersection Number.- 1.2. Additivity.- 1.3. Invariance Under Linear Equivalence.- 1.4. The General Definition of Intersection Number.- Exercises to §1.- 2. Applications of Intersection Numbers.- 2.1. Bézout's Theorem in Projective and Multiprojective Space.- 2.2. Varieties over the Reals.- 2.3. The Genus of a Nonsingular Curve on a Surface.- 2.4. The Riemann-Roch Inequality on a Surface.- 2.5. The Nonsingular Cubic Surface.- 2.6. The Ring of Cycle Classes.- Exercises to §2.- 3. Birational Maps of Surfaces.- 3.1. Blowups of Surfaces.- 3.2. Some Intersection Numbers.- 3.3. Resolution of Indeterminacy.- 3.4. Factorisation as a Chain of Blowups.- 3.5. Remarks and Examples.- Exercises to §3.- 4. Singularities.- 4.1. Singular Points of a Curve.- 4.2. Surface Singularities.- 4.3. Du Val Singularities.- 4.4. Degeneration of Curves.- Exercises to §4.- Algebraic Appendx.- References.