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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. $\endgroup$ – Tom Leinster Oct 25 '09 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. $\endgroup$ – Ilya Nikokoshev Oct 25 '09 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. $\endgroup$ – Tom Leinster Oct 27 '09 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. $\endgroup$ – Harry Gindi Dec 17 '09 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. $\endgroup$ – darij grinberg Jun 1 '10 at 20:54

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I've found something extraordinary and of equally extrordinary 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, Micheal 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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    $\begingroup$ Dear Andrew, please put spaces after your punctuation. $\endgroup$ – Anweshi Jul 25 '10 at 7:52
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    $\begingroup$ @Anweshi: Andrew has stated before that this is due to some typesetting bug on his end. @Andrew: I took this class for most of the semester. The lecture notes were actually scribed by the students, so caveat emptor. $\endgroup$ – Qiaochu Yuan Jul 25 '10 at 9:14
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    $\begingroup$ For what it's worth, I don't really believe that there's a bug causing these issues. Looking at Andrew L's posts over time, there has been a gradual improvement in the use of correct punctuation. I find it hard to imagine a software issue with these effects. $\endgroup$ – Scott Morrison Jul 25 '10 at 18:36
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    $\begingroup$ @To All Above: To be honest,it's really a little of both. Sometimes,the LaTeX doesn't cooperate and I'm still learning how to use it. Then again,I'm not really a stickler for grammatical etiquitte.The second observation is really not professional on my part and I'm seriously trying to make an effort to improve. @Qiaochu This is the fundamental problem with all free lecture note sources. Artin's algebra book,however,began life that way and that worked out pretty well.Let's hope he finds time to edit them and post corrected versions. $\endgroup$ – The Mathemagician Jul 25 '10 at 19:19
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    $\begingroup$ @Andrew L: It is not true that I will your throw writings away if you miss a comma. I edited a good number of your posts which were almost impossible to decipher and even requested you to write better. For instance the comments left at mathoverflow.net/questions/32736 . If you want others to read your stuff, please at least put in some effort to reduce typographical unpleasantness. If you didn't know English and if you were a mathematical genius with linguistic difficulties, then this could be tolerated. But Scott's comment above indicates that you actually know how to write properly; $\endgroup$ – Anweshi Jul 26 '10 at 21:18
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Manin's lectures on algebraic geometry that were recently translated into English could be helpful too.

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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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  • $\begingroup$ do you know group law on a elliptic curve? $\endgroup$ – Koushik Jun 6 '14 at 9:05
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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... $\endgroup$ – Thierry Zell Sep 28 '11 at 16:00
  • $\begingroup$ @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. $\endgroup$ – François Brunault Sep 28 '11 at 16:59
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Another nice introductory AG book that, I believe, was not mentioned here yet is Hassett.

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