Algebraic Geometry: A First Course

Joe Harris (auth.)9780387977164, 0387977163

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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Table of contents :
Front Matter….Pages i-xix
Front Matter….Pages 1-1
Affine and Projective Varieties….Pages 3-16
Regular Functions and Maps….Pages 17-31
Cones, Projections, and More About Products….Pages 32-40
Families and Parameter Spaces….Pages 41-47
Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz….Pages 48-62
Grassmannians and Related Varieties….Pages 63-71
Rational Functions and Rational Maps….Pages 72-87
More Examples….Pages 88-97
Determinantal Varieties….Pages 98-113
Algebraic Groups….Pages 114-129
Front Matter….Pages 131-131
Definitions of Dimension and Elementary Examples….Pages 133-150
More Dimension Computations….Pages 151-162
Hilbert Polynomials….Pages 163-173
Smoothness and Tangent Spaces….Pages 174-185
Gauss Maps, Tangential and Dual Varieties….Pages 186-199
Tangent Spaces to Grassmannians….Pages 200-210
Further Topics Involving Smoothness and Tangent Spaces….Pages 211-223
Degree….Pages 224-238
Further Examples and Applications of Degree….Pages 239-250
Singular Points and Tangent Cones….Pages 251-265
Front Matter….Pages 131-131
Parameter Spaces and Moduli Spaces….Pages 266-281
Quadrics….Pages 282-307
Back Matter….Pages 308-330