Algebraic Geometry

1. Front Matter

   Pages i-xix

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2. Examples of Varieties and Maps

   1. Front Matter

      Pages 1-1

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   2. Affine and Projective Varieties

      • Joe Harris

      Pages 3-16

   3. Regular Functions and Maps

      • Joe Harris

      Pages 17-31

   4. Cones, Projections, and More About Products

      • Joe Harris

      Pages 32-40

   5. Families and Parameter Spaces

      • Joe Harris

      Pages 41-47

   6. Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz

      • Joe Harris

      Pages 48-62

   7. Grassmannians and Related Varieties

      • Joe Harris

      Pages 63-71

   8. Rational Functions and Rational Maps

      • Joe Harris

      Pages 72-87

   9. More Examples

      • Joe Harris

      Pages 88-97

   10. Determinantal Varieties

       • Joe Harris

       Pages 98-113

   11. Algebraic Groups

       • Joe Harris

       Pages 114-129

3. Attributes of Varieties

   1. Front Matter

      Pages 131-131

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   2. Definitions of Dimension and Elementary Examples

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      Pages 133-150

   3. More Dimension Computations

      • Joe Harris

      Pages 151-162

   4. Hilbert Polynomials

      • Joe Harris

      Pages 163-173

   5. Smoothness and Tangent Spaces

      • Joe Harris

      Pages 174-185

   6. Gauss Maps, Tangential and Dual Varieties

      • Joe Harris

      Pages 186-199

   7. Tangent Spaces to Grassmannians

      • Joe Harris

      Pages 200-210

   8. Further Topics Involving Smoothness and Tangent Spaces

      • Joe Harris

      Pages 211-223

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.

• Dimension
• Grad
• algebraic geometry
• algebraic group

J. Harris

Algebraic Geometry

A First Course

"This book succeeds brilliantly by concentrating on a number of core topics (the rational normal curve, Veronese and Segre maps, quadrics, projections, Grassmannians, scrolls, Fano varieties, etc.) and by treating them in a hugely rich and varied way. The author ensures that the reader will learn a large amount of classical material and perhaps more importantly, will also learn that there is no one approach to the subject. The essence lies in the range and interplay of possible approaches. The author is to be congratulated on a work of deep and enthusiastic scholarship."—MATHEMATICAL REVIEWS

• Department of Mathematics, Harvard University, Cambridge, USA

  Joe Harris

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