An Invitation to Algebraic Geometry

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Karen E. Smith
Springer Science & Business Media, Oct 31, 2000 - Mathematics - 155 pages
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The aim of this book is to describe the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Few algebraic prerequisites are presumed beyond a basic course in linear algebra.
  

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Contents

Affine Algebraic Varieties
1
11 Definition and Examples
2
12 The Zariski Topology
6
13 Morphisms of Affine Algebraic Varieties
9
14 Dimension
11
Algebraic Foundations
15
22 Hilberts Basis Theorem
18
23 Hilberts Nullstellensatz
20
56 The Hilbert Function
81
Smoothness
85
62 Smooth Points
92
63 Smoothness in Families
96
64 Bertinis Theorem
99
65 The Gauss Mapping
102
Birational Geometry
105
72 Rational Maps
112

24 The Coordinate Ring
23
25 The Equivalence of Algebra and Geometry
26
26 The Spectrum of a Ring
30
Projective Varieties
33
32 Projective Varieties
37
33 The Projective Closure of an Affine Variety
41
34 Morphisms of Projective Varieties
44
35 Automorphisms of Projective Space
47
QuasiProjective Varieties
51
42 A Basis for the Zariski Topology
55
43 Regular Functions
56
Classical Constructions
63
52 Five Points Determine a Conic
65
53 The Segre Map and Products of Varieties
67
54 Grassmannians
71
55 Degree
74
73 Birational Equivalence
114
74 Blowing Up Along an Ideal
115
75 Hypersurfaces
119
76 The Classification Problems
120
Maps to Projective Space
123
81 Embedding a Smooth Curve in Three Space
124
82 Vector Bundles and Line Bundles
127
83 The Sections of a Vector Bundle
129
84 Examples of Vector Bundles
131
85 Line Bundles and Rational Maps
136
86 Very Ample Line Bundles
141
Sheaves and Abstract Algebraic Varieties
145
A2 Abstract Algebraic Varieties
150
References
153
Index
157
Copyright

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About the author (2000)

Karen E. Smith is an Associate Professor of Mathematics at the University of Michigan.

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