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Algebraic Varieties

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ISBN-10: 0521426138

ISBN-13: 9780521426138

Edition: 1993

Authors: George R. Kempf, J. W. S. Cassels, N. J. Hitchin

List price: $61.99
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Book details

List price: $61.99
Copyright year: 1993
Publisher: Cambridge University Press
Publication date: 9/9/1993
Binding: Paperback
Pages: 176
Size: 6.25" wide x 9.25" long x 0.50" tall
Weight: 0.550
Language: English

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