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Algebraic Varieties : London Mathematical Society Lecture Note Series - G. Kempf

Algebraic Varieties

London Mathematical Society Lecture Note Series

By: G. Kempf


Published: 8th November 1993
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In this book Professor Kempf gives an introduction to the theory of algebraic functions on varieties from a sheaf theoretic standpoint. By taking this view he is able to give a clean and lucid account of the subject which will be easily accessible to all newcomers to algebraic varieties, graduate students or experts from other fields alike. Anyone who goes on to study schemes will find that this book is an ideal preparatory text.

"...the material is presented in a nice way; the author uses geometric language...recommended for graduate students who are interested in algebraic geometry." Gerhard Pfister, Mathematical Reviews "Excellent background for study of schemes." American Mathematical Monthly

Introductionp. ix
Algebraic varieties: definition and existencep. 1
Spaces with functionsp. 1
Varietiesp. 2
The existence of affine varietiesp. 4
The nullstellensatzp. 5
The rest of the proof of existence of affine varieties / subvarietiesp. 8
A[superscript n] and P[superscript n]p. 10
Determinantal varietiesp. 11
The preparation lemma and some consequencesp. 13
The lemmap. 13
The Hilbert basis theoremp. 15
Irreducible componentsp. 16
Affine and finite morphismsp. 18
Dimensionp. 20
Hypersurfaces and the principal ideal theoremp. 21
Products; separated and complete varietiesp. 25
Productsp. 25
Products of projective varietiesp. 27
Graphs of morphisms and separatednessp. 28
Algebraic groupsp. 30
Cones and projective varietiesp. 31
A little more dimension theoryp. 32
Complete varietiesp. 33
Chow's lemmap. 34
The group law on an elliptic curvep. 35
Blown up A[superscript n] at the originp. 36
Sheavesp. 38
The definition of presheaves and sheavesp. 38
The construction of sheavesp. 42
Abelian sheaves and flabby sheavesp. 46
Direct limits of sheavesp. 50
Sheaves in algebraic geometryp. 54
Sheaves of rings and modulesp. 54
Quasi-coherent sheaves on affine varietiesp. 56
Coherent sheavesp. 58
Quasi-coherent sheaves on projective varietiesp. 61
Invertible sheavesp. 62
Operations on sheaves that change spacesp. 65
Morphisms to projective space and affine morphismsp. 68
Smooth varieties and morphismsp. 70
The Zariski cotangent space and smoothnessp. 70
Tangent conesp. 72
The sheaf of differentialsp. 75
Morphismsp. 80
The construction of affine morphisms and normalizationp. 82
Bertini's theoremp. 83
Curvesp. 85
Introduction to curvesp. 85
Valuation criterionsp. 87
The construction of all smooth curvesp. 88
Coherent sheaves on smooth curvesp. 90
Morphisms between smooth complete curvesp. 92
Special morphisms between curvesp. 94
Principal parts and the Cousin problemp. 96
Cohomology and the Riemann-Roch theoremp. 98
The definition of cohomologyp. 98
Cohomology of affinesp. 100
Higher direct imagesp. 102
Beginning the study of the cohomology of curvesp. 104
The Riemann-Roch theoremp. 106
First applications of the Riemann-Roch theoremp. 108
Residues and the trace homomorphismp. 110
General cohomologyp. 113
The cohomology of A[superscript n] - 0 and P[superscript n]p. 113
Cech cohomology and the Kunneth formulap. 114
Cohomology of projective varietiesp. 116
The direct images of flat sheavesp. 118
Families of cohomology groupsp. 120
Applicationsp. 124
Embedding in projective spacep. 124
Cohomological characterization of affine varietiesp. 125
Computing the genus of a plane curve and Bezout's theoremp. 126
Elliptic curvesp. 128
Locally free coherent sheaves on P[superscript 1]p. 129
Regularity in codimension onep. 130
One dimensional algebraic groupsp. 131
Correspondencesp. 132
The Reimann-Roch theorem for surfacesp. 139
Appendixp. 139
Localizationp. 141
Direct limitsp. 143
Eigenvectorsp. 144
Bibliographyp. 146
Glossary of notationp. 149
Indexp. 155
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521426138
ISBN-10: 0521426138
Series: London Mathematical Society Lecture Note Series
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 176
Published: 8th November 1993
Country of Publication: GB
Dimensions (cm): 23.5 x 15.88  x 1.27
Weight (kg): 0.25