| Introduction | p. ix |
| Algebraic varieties: definition and existence | p. 1 |
| Spaces with functions | p. 1 |
| Varieties | p. 2 |
| The existence of affine varieties | p. 4 |
| The nullstellensatz | p. 5 |
| The rest of the proof of existence of affine varieties / subvarieties | p. 8 |
| A[superscript n] and P[superscript n] | p. 10 |
| Determinantal varieties | p. 11 |
| The preparation lemma and some consequences | p. 13 |
| The lemma | p. 13 |
| The Hilbert basis theorem | p. 15 |
| Irreducible components | p. 16 |
| Affine and finite morphisms | p. 18 |
| Dimension | p. 20 |
| Hypersurfaces and the principal ideal theorem | p. 21 |
| Products; separated and complete varieties | p. 25 |
| Products | p. 25 |
| Products of projective varieties | p. 27 |
| Graphs of morphisms and separatedness | p. 28 |
| Algebraic groups | p. 30 |
| Cones and projective varieties | p. 31 |
| A little more dimension theory | p. 32 |
| Complete varieties | p. 33 |
| Chow's lemma | p. 34 |
| The group law on an elliptic curve | p. 35 |
| Blown up A[superscript n] at the origin | p. 36 |
| Sheaves | p. 38 |
| The definition of presheaves and sheaves | p. 38 |
| The construction of sheaves | p. 42 |
| Abelian sheaves and flabby sheaves | p. 46 |
| Direct limits of sheaves | p. 50 |
| Sheaves in algebraic geometry | p. 54 |
| Sheaves of rings and modules | p. 54 |
| Quasi-coherent sheaves on affine varieties | p. 56 |
| Coherent sheaves | p. 58 |
| Quasi-coherent sheaves on projective varieties | p. 61 |
| Invertible sheaves | p. 62 |
| Operations on sheaves that change spaces | p. 65 |
| Morphisms to projective space and affine morphisms | p. 68 |
| Smooth varieties and morphisms | p. 70 |
| The Zariski cotangent space and smoothness | p. 70 |
| Tangent cones | p. 72 |
| The sheaf of differentials | p. 75 |
| Morphisms | p. 80 |
| The construction of affine morphisms and normalization | p. 82 |
| Bertini's theorem | p. 83 |
| Curves | p. 85 |
| Introduction to curves | p. 85 |
| Valuation criterions | p. 87 |
| The construction of all smooth curves | p. 88 |
| Coherent sheaves on smooth curves | p. 90 |
| Morphisms between smooth complete curves | p. 92 |
| Special morphisms between curves | p. 94 |
| Principal parts and the Cousin problem | p. 96 |
| Cohomology and the Riemann-Roch theorem | p. 98 |
| The definition of cohomology | p. 98 |
| Cohomology of affines | p. 100 |
| Higher direct images | p. 102 |
| Beginning the study of the cohomology of curves | p. 104 |
| The Riemann-Roch theorem | p. 106 |
| First applications of the Riemann-Roch theorem | p. 108 |
| Residues and the trace homomorphism | p. 110 |
| General cohomology | p. 113 |
| The cohomology of A[superscript n] - 0 and P[superscript n] | p. 113 |
| Cech cohomology and the Kunneth formula | p. 114 |
| Cohomology of projective varieties | p. 116 |
| The direct images of flat sheaves | p. 118 |
| Families of cohomology groups | p. 120 |
| Applications | p. 124 |
| Embedding in projective space | p. 124 |
| Cohomological characterization of affine varieties | p. 125 |
| Computing the genus of a plane curve and Bezout's theorem | p. 126 |
| Elliptic curves | p. 128 |
| Locally free coherent sheaves on P[superscript 1] | p. 129 |
| Regularity in codimension one | p. 130 |
| One dimensional algebraic groups | p. 131 |
| Correspondences | p. 132 |
| The Reimann-Roch theorem for surfaces | p. 139 |
| Appendix | p. 139 |
| Localization | p. 141 |
| Direct limits | p. 143 |
| Eigenvectors | p. 144 |
| Bibliography | p. 146 |
| Glossary of notation | p. 149 |
| Index | p. 155 |
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