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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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    $\begingroup$ 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. $\endgroup$ Oct 25, 2009 at 16:02
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    $\begingroup$ 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. $\endgroup$ Oct 25, 2009 at 21:52
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    $\begingroup$ 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. $\endgroup$ Oct 27, 2009 at 4:50
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    $\begingroup$ 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. $\endgroup$ Dec 17, 2009 at 3:50
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    $\begingroup$ -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. $\endgroup$ Jun 1, 2010 at 20:54

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Read Math Overflow.

Whereas it is actually not quite a textbook, it is becoming a very popular reference. In recent talks it was even used as the almost exclusively!

And indeed, there are a lot of high quality 'articles', and often you can find alternative approaches to a theory or a problem, which are more suitable for you. In addition, you can actually ask questions (a feature thoroughly missed in e.g. Hartshorne's book).

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I've found something extraordinary and of equally extraordinary pedigree online recently. I mentioned it briefly in response to R. Vakil's question about the best way to introduce schemes to students. But this question is really where it belongs and I hope word of it spreads far and wide from here.

Last fall at MIT, Michael Artin taught an introductory course in algebraic geometry that required only a year of basic algebra at the level of his textbook. The official text was William Fulton's Algebraic Curves, but Artin also wrote an extensive set of lecture notes and exercise sets. I found them quite wonderful and very much in the spirit of his classic textbook. (By the way, simply can't wait for the second edition.)

Not only has he posted these notes for download, he's asked anyone working through them to email him any errors found and suggestions for improvements. All the course materials can be found at the MIT webpage. I've also posted the link at MathOnline, of course.

I don't know if most of the hardcore algebraic geometers here would recommend these materials for a beginning course. But for any student not looking to specialize in AG, I can't think of a better source to begin with. That's just my opinion. But it certainly belongs as a possible response to this question. Then again, it may be too softball for the experts, particularly those of the Grothendieck school.

Here's keeping our fingers crossed that this is the beginning of the gestation of a full blown text on the subject by Artin.

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Manin's lectures on algebraic geometry that were recently translated into English could be helpful too.

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I think it's hard to say which one is the best, but for my own experience, I got into this area pretty much by reading most of this "3264 & all that" book, and completing almost all exercises (this is crucial!). It is said to be on intersection theory, but when I worked through it, I learned many other perspectives and came up with lots of concrete questions as well. I strongly recommend this book (again, the crucial point is doing exercises).

But of course I think it would be good to not just stick on one book. For example, Hartshorne definitely has a very quick and useful intro in cohomology, but for the part "higher direct images" I think 3264 is better. Also Beauville's surface book is of course good intro to surfaces, but for discussion of ruled surfaces, I think Hartshorne is actuallty better...

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Another nice introductory AG book that, I believe, was not mentioned here yet is Hassett.

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About Hartshorne and Griffiths, I think a comparison between the two texts is misleading. The first is a introduction to the "Grothendieck yoga" where geometrical classical ideas are "immersed" in the larger but abstract mathematical world of schemes. But also if the complex differential manifold style of Griffiths is "more concrete" is very different from the "Algebraic Geometry" idea, also if it is a deep study of it.

As a categorist I love Grothendieck, but I find Grothendieck's work fantastic in itself (like a music), also if I never understand anything about Geometry while studying or reading EGA or some SGA.

For understanding 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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    $\begingroup$ 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... $\endgroup$ Sep 28, 2011 at 16:00
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    $\begingroup$ @unknown : Do you mean Fulton's book Algebraic curves ? It has already been mentioned in several answers. I agree this is a very good book. $\endgroup$ Sep 28, 2011 at 16:59
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