Algebraic Geometry (book)

Influential textbook about algebraic geometry, written by Robin Hartshorne

Algebraic Geometry

First edition

Author| Robin Hartshorne
Language| English
Subject| Algebraic geometry
Genre| Textbook
Published| 1977

Algebraic Geometry is an algebraic geometry textbook written by Robin Hartshorne and published by Springer-Verlag in 1977.[1]

## Importance

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It was the first extended treatment of scheme theory written as a text intended to be accessible to graduate students, and is considered to be the standard reference.[2][3][4] This book was cited when Hartshorne was awarded the Leroy P. Steele Prize for mathematical exposition in 1979.

## Contents

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The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties over algebraically closed fields. This chapter uses many classical results in commutative algebra, including Hilbert's Nullstellensatz, with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart of the book. The last two chapters, "Curves" and "Surfaces", respectively explore the geometry of 1- and 2-dimensional objects, using the tools developed in the chapters 2 and 3.

## Notes

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1. ^ MathSciNet lists more than 2500 citations of this book.
2. ^ Reid, Miles (1989). Undergraduate Algebraic Geometry. Cambridge University Press. p. 8. This is the professional’s handbook.
3. ^ Harris, Joe (1992). Algebraic Geometry: A First Course. Springer-Verlag. p. xii. Hartshorne's classic book stands out as the canonical reference.
4. ^ Green, Mark (2002). "Review: An Invitation to Algebraic Geometry by K Smith; L Kahanpää; P Kekäläinen; W Traves". Amer. Math. Monthly. 109 (7): 675–678. ...has endured as the best one-volume treatment of this essential set of tools. Everyone in algebraic geometry eventually studies this book.

## References

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* Hartshorne, Robin (1977). Algebraic Geometry. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4757-3849-0. ISBN 978-0-387-90244-9. MR 0463157. Zbl 0367.14001.
* Shatz, Stephen S. (1979), "Review: Robin Hartshorne, Algebraic geometry", Bull. Amer. Math. Soc. (N.S.), 1 (3): 553–560, doi:10.1090/S0273-0979-1979-14618-4