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1 Affine Algebraic Varieties 1

1.1 Definition and Examples 2

1.2 The Zariski Topology 6

1.3 Morphisms of Affine Algebraic Varieties 9

1.4 Dimension 11

2 Algebraic Foundations 15

2.1 A Quick Review of Commutative Ring Theory 15

2.2 Hilbert's Basis Theorem 18

2.3 Hilbert's Nullstellensatz 20

2.4 The Coordinate Ring 23

2.5 The Equivalence of Algebra and Geometry 26

2.6 The Spectrum of a Ring 30

3 Projective Varieties 33

3.1 Projective Space 33

3.2 Projective Varieties 37

3.3 The Projective Closure of an Affine Variety 41

3.4 Morphisms of Projective Varieties 44

3.5 Automorphisms of Projective Space 47

4 Quasi-Projective Varieties 51

4.1 Quasi-Projective Varieties 51

4.2 A Basis for the Zariski Topology 55

4.3 Regular Functions 56

5 Classical Constructions 63

5.1 Veronese Maps 63

5.2 Five Points Determine a Conic 65

5.3 The Segre Map and Products of Varieties 67

5.4 Grassmannians 71

5.5 Degree 74

5.6 The Hilbert Function 81

6 Smoothness 85

6.1 The Tangent Space at a Point 85

6.2 Smooth Points 92

6.3 Smoothness in Families 96

6.4 Bertini's Theorem 99

6.5 The Gauss Mapping 102

7 Birational Geometry 105

7.1 Resolution of Singularities 105

7.2 Rational Maps 112

7.3 Birational Equivalence 114

7.4 Blowing Up Along an Ideal 115

7.5 Hypersurfaces 119

7.6 The Classification Problems 120

8 Maps to Projective Space 123

8.1 Embedding a Smooth Curve in Three-Space 124

8.2 Vector Bundles and Line Bundles 127

8.3 The Sections of a Vector Bundle 129

8.4 Examples of Vector Bundles 131

8.5 Line Bundles and Rational Maps 136

8.6 Very Ample Line Bundles 141

A Sheaves and Abstract Algebraic Varieties 145

A.1 Sheaves 145

A.2 Abstract Algebraic Varieties 150

References 153

Index 157