Algebraic Geometry: A First Course

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Springer Science & Business Media, 11 de nov. 2013 - 330 pàgines
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This book is based on one-semester courses given at Harvard in 1984, at Brown in 1985, and at Harvard in 1988. It is intended to be, as the title suggests, a first introduction to the subject. Even so, a few words are in order about the purposes of the book. Algebraic geometry has developed tremendously over the last century. During the 19th century, the subject was practiced on a relatively concrete, down-to-earth level; the main objects of study were projective varieties, and the techniques for the most part were grounded in geometric constructions. This approach flourished during the middle of the century and reached its culmination in the work of the Italian school around the end of the 19th and the beginning of the 20th centuries. Ultimately, the subject was pushed beyond the limits of its foundations: by the end of its period the Italian school had progressed to the point where the language and techniques of the subject could no longer serve to express or carry out the ideas of its best practitioners.
 

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Continguts

A Note on Dimension Smoothness and Degree
16
LECTURE
17
Projective Varieties
20
LECTURE 3
32
More Projections
38
Parameter Spaces of Hypersurfaces
44
Ideals of Projective Varieties
50
A Little Algebra
57
LECTURE 16
200
The Resolution of the Generic Determinantal Variety
206
Further Topics Involving Smoothness and Tangent Spaces
211
Blowups Nash Blowups and the Resolution of Singularities
219
Bézouts Theorem
227
LECTURE 19
239
Degrees of Some Grassmannians
245
LECTURE 20
251

LECTURE 6
63
LECTURE 7
72
Unirationality
87
More General Determinantal Varieties
111
Quotients
123
ATTRIBUTES OF VARIETIES
130
Immediate Examples
138
LECTURE 12
151
Group Actions
161
Syzygies
168
LECTURE 14
174
Projective Tangent Spaces
181
Multiplicity
258
Resolution of Singularities for Curves
264
Hilbert Varieties
273
Curves of Degree 2
275
LECTURE 22
282
Linear Spaces on Quadrics
289
Families of Quadrics
295
Pencils of Quadrics
301
Hints for Selected Exercises
308
References
314
Copyright

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