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Algebraic Geometry (book)
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 First edition | |
| Author | Robin Hartshorne |
|---|---|
| Language | English |
| Series | Graduate Texts in Mathematics; Vol 52 |
| Subject | Algebraic geometry |
| Genre | Textbook |
| Published | 1977 |
| Pages | XVI + 496 |
| ISBN | 978-0-387-90244-9 |
| Dewey Decimal | 516.35 |
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scheme theory like Macdonald's Algebraic Geometry: Introduction to Schemes (1968) and Mumford's widely circulated but unpublished Red Book (1967) were available earlier, Hartshorne's Algebraic Geometry was the first comprehensive treatment of scheme theory written as a text intended to be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 have been considered by some to be a rite-of-passage for serious students and future researchers 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", gives a brief, formal introduction to the classical algebraic geometry of varieties over algebraically closed fields. This chapter recalls many classical results from commutative algebra (e.g., Hilbert's Nullstellensatz), with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel cited as references for proofs. 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 (1988), p. 8, "This is the professional's handbook".
- ↑ Harris (1992), p. xii, "Hartshorne's classic book stands out as the canonical reference".
- ↑ Green (2002), "[Hartshorne] 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
- Green, Mark (2002). "Review: An Invitation to Algebraic Geometry by K Smith; L Kahanpää; P Kekäläinen; W Traves". American Mathematical Monthly. 109 (7): 675–678. JSTOR 3072450.
- Harris, Joe (1992). Algebraic Geometry: A First Course. Graduate Texts in Mathematics. Vol. 133. New York: Springer. ISBN 978-0-387-97716-4.
- Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. New York: Springer. ISBN 978-0-387-90244-9. MR 0463157. Zbl 0367.14001.
- Reid, Miles (1988). Undergraduate Algebraic Geometry. London Mathematical Society Student Texts. Vol. 12. Cambridge University Press. ISBN 978-0-521-35559-9.
- Shatz, Stephen S. (1979). "Review: Robin Hartshorne, Algebraic geometry". Bulletin of the American Mathematical Society. 1 (3): 553–560. doi:10.1090/S0273-0979-1979-14618-4.