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I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc.

One suggestion per answer please. Also, please include an explanation of why you like the book, or what makes it unique or useful.

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Since I'm not an algebraic geometer, I don't know whether I'm qualified to comment. But if I am, I've got to disagree about Hartshorne. Every time I open my copy, I think "God, this makes algebraic geometry look unappetizing". Maybe if I worked through it systematically I'd like it. But as a reference for a non-expert, it's pretty off-putting, I find. –  Tom Leinster Oct 25 '09 at 16:02
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Let me present my perspective on "Hartshorne is best issue". It's certainly very systematic with lots of exercises and a wonderful reference book, but it's only useful to people who somehow got the motivation to study abstract algebraic geometry, not as the first book. –  Ilya Nikokoshev Oct 25 '09 at 21:52
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I can believe it's a wonderful reference, but I've found it unsatisfying at the conceptual level. Two examples: 1. He never mentions that the category of affine schemes is dual to the category of rings, as far as I can see. I'd expect to see that in huge letters near the definition of scheme. How could you miss that out? 2. He puts the condition "F(emptyset) is trivial" into the definition of presheaf, when really it belongs in the definition of sheaf. That's a small thing, but hinders the reader from getting a good understanding of these important concepts. –  Tom Leinster Oct 27 '09 at 4:50
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Even worse than that, his construction of the structure sheaf basically rigs it so the stalks are the localizations at the primes, and doesn't even try to explain what's going on. There's no motivation, and it's not even described in a theorem or definition or theorem/definition. The reduced induced closed subscheme is introduced in an example, etc. It's not a book that you can read, it's a book that you have to work through. –  Harry Gindi Dec 17 '09 at 3:50
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-1 for "I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best." It may be a decent reference that one takes with oneself on a journey for the case one should need some result, but as a textbook it is useless. –  darij grinberg Jun 1 '10 at 20:54

32 Answers 32

About HArtshorne and Griffith, I think a comparison between the two texts is misleading. The first is a introduction to the "Grothendieck ioga" where geometrical classical idea are "immersed" in the more large but abstract mathematical world of schemas. But also if the Complex differential manifold style of Griffith is "more concrete" is very different from the "Algebraic Geometry" idea, also if it is a deep study of it.

As categorist I love Grothendiek, but I find Grothendieck work fantastic in itself (like a music), also if I never understand nothing about Geometry Geometry while studing or readind EGA or some SGA.

FOr understant what is "ALgebraic Geometry" I had to read this

Beltrametti-Carletti-Gallarati-Monti Bragadin, Letture su curve, superficie e varietà proiettive speciali, Boringhieri

Beltrametti-Carletti-Gallarati-Monti Bragadin, Lezioni di geometria analitica e proiettiva, Boringhieri

(sorry, these are in Italian)

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The word "best" is relative. If you have a strong background in commutative algebra and have had considerable exposure to algebraic geometry I would say Hartshorne would suit you. But for an introductory graduate text, I don't think so. We're using Fulton. Organization and exposition is okay, and the discussion is not as "hardcore" as that of Hartshorne. I'm surprised it didn't show up from those of you who posted here.

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Actually, Eisenbud is on the record as saying that he wrote his tome as (to paraphrase) "the book one should have read before tackling Hartshorne's". Given the size of Eisenbud's book... –  Thierry Zell Sep 28 '11 at 16:00

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