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| Introduction | |
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| Algebraic varieties: definition and existence | |
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| Spaces with functions | |
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| Varieties | |
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| The existence of affine varieties | |
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| The nullstellensatz | |
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| The rest of the proof of existence of affine varieties / subvarieties | |
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| A[superscript n] and P[superscript n] | |
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| Determinantal varieties | |
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| The preparation lemma and some consequences | |
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| The lemma | |
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| The Hilbert basis theorem | |
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| Irreducible components | |
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| Affine and finite morphisms | |
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| Dimension | |
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| Hypersurfaces and the principal ideal theorem | |
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| Products; separated and complete varieties | |
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| Products | |
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| Products of projective varieties | |
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| Graphs of morphisms and separatedness | |
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| Algebraic groups | |
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| Cones and projective varieties | |
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| A little more dimension theory | |
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| Complete varieties | |
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| Chow's lemma | |
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| The group law on an elliptic curve | |
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| Blown up A[superscript n] at the origin | |
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| Sheaves | |
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| The definition of presheaves and sheaves | |
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| The construction of sheaves | |
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| Abelian sheaves and flabby sheaves | |
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| Direct limits of sheaves | |
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| Sheaves in algebraic geometry | |
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| Sheaves of rings and modules | |
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| Quasi-coherent sheaves on affine varieties | |
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| Coherent sheaves | |
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| Quasi-coherent sheaves on projective varieties | |
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| Invertible sheaves | |
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| Operations on sheaves that change spaces | |
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| Morphisms to projective space and affine morphisms | |
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| Smooth varieties and morphisms | |
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| The Zariski cotangent space and smoothness | |
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| Tangent cones | |
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| The sheaf of differentials | |
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| Morphisms | |
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| The construction of affine morphisms and normalization | |
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| Bertini's theorem | |
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| Curves | |
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| Introduction to curves | |
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| Valuation criterions | |
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| The construction of all smooth curves | |
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| Coherent sheaves on smooth curves | |
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| Morphisms between smooth complete curves | |
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| Special morphisms between curves | |
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| Principal parts and the Cousin problem | |
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| Cohomology and the Riemann-Roch theorem | |
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| The definition of cohomology | |
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| Cohomology of affines | |
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| Higher direct images | |
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| Beginning the study of the cohomology of curves | |
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| The Riemann-Roch theorem | |
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| First applications of the Riemann-Roch theorem | |
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| Residues and the trace homomorphism | |
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| General cohomology | |
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| The cohomology of A[superscript n] - 0 and P[superscript n] | |
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| Cech cohomology and the Kunneth formula | |
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| Cohomology of projective varieties | |
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| The direct images of flat sheaves | |
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| Families of cohomology groups | |
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| Applications | |
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| Embedding in projective space | |
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| Cohomological characterization of affine varieties | |
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| Computing the genus of a plane curve and Bezout's theorem | |
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| Elliptic curves | |
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| Locally free coherent sheaves on P[superscript 1] | |
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| Regularity in codimension one | |
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| One dimensional algebraic groups | |
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| Correspondences | |
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| The Reimann-Roch theorem for surfaces | |
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| Appendix | |
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| Localization | |
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| Direct limits | |
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| Eigenvectors | |
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| Bibliography | |
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| Glossary of notation | |
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| Index | |