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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 | |