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Notes for the Second PrintingPrefaceAcknowledgmentsIndex of Notation1 Affine Algebraic Varieties　1.1 Definition and Examples　1.2 The Zariski Topology　1.3 Morphisms of Affine Algebraic Varieties　1.4 Dimension2 Algebraic Foundations　2.1 A Quick Review of Commutative Ring Theory　2.2 Hilbert's Basis Theorem　2.3 Hilbert's NuUstellensatz　2.4 The Coordinate Ring　2.5 The Equivalence of Algebra and Geometry　2.6 The Spectrum of a Ring3 Projective Varieties　3.1 Projective Space　3.2 Projective Varieties　3.3 The Projective Closure of an Affine Variety　3.4 Morphisms of Projective Varieties　3.5 Automorphisms of Projective Space4 Quasi-Projective Varieties　4.1 Quasi-Projective Varieties　4.2 A Basis for the Zariski Topology　4.3 Regular Functions5 Classical Constructions　5.1 Veronese Maps　5.2 Five Points Determine a Conic　5.3 The Segre Map and Products of Varieties　5.4 Grassmannians　5.5 Degree　5.6 The Hilbert Function6 Smoothness　6.1 The Tangent Space at a Point　6.2 Smooth Points　6.3 Smoothness in Families　6.4 Bertini's Theorem　6.5 The Gauss Mapping7 Birational Geometry　7.1 Resolution of Singularities　7.2 Rational Maps　7.3 Birational Equivalence　7.4 Blowing Up Along an Ideal　7.5 Hypersurfaces　7.6 The Classification Problems8 Maps to Projective Space　8.1 Embedding a Smooth Curve in Three-Space　8.2 Vector Bundles and Line Bundles　8.3 The Sections of a Vector Bundle　8.4 Examples of Vector Bundles　8.5 Line Bundles and Rational Maps　8.6 Very Ample Line BundlesA Sheaves and Abstract Algebraic Varieties　A.1 Sheaves　A.2 Abstract Algebraic VarietiesReferencesIndex