Algebraic Geometry (book)
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
Although shorter synopses introducing scheme theory like Macdonald's Algebraic Geometry: Introduction to Schemes (1968) and Mumford's widely circulated but unpublished Red Book (1967) were available earlier, Hartshorne's Algebraic Geometry was the first comprehensive treatment of scheme theory written as a text intended to be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] Even so, it is considered to be notoriously difficult for beginners; reading it and working through the challenging exercises has been considered by some to be a rite-of-passage for students of algebraic geometry.[5] This book was cited when Hartshorne was awarded the Leroy P. Steele Prize for mathematical exposition in 1979.
Contents
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
- ^ MathSciNet lists more than 2500 citations of this book.
- ^ Reid, Miles (1989). Undergraduate Algebraic Geometry. Cambridge University Press. p. 8. This is the professional’s handbook.
- ^ Harris, Joe (1992). Algebraic Geometry: A First Course. Springer-Verlag. p. xii. Hartshorne's classic book stands out as the canonical reference.
- ^ 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.
- ^ "Terence Tao's blog: Learn and relearn your field (comment by Matthew Emerton)". What's new. 2007-05-06. Retrieved 2026-03-15.
Most students in algebraic geometry, of all flavours, go through the rite of passage known as "Hartshorne": reading Hartshorne's book, especially chapters 2 and 3, and solving vast numbers of exercises. It is more or less impossible, and in any case probably unwise, to avoid doing this. And once you have solved many/most of the Hartshorne problems, you should have some baseline confidence in algebraic geometry, scheme theory, and cohomology.
References
- 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