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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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 Community wiki this? – Kevin Lin Oct 25 at 13:47
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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 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 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 at 4:50
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. – fpqc Dec 17 at 3:50
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21 Answers

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I think the best "textbook" is Ravi Vakil's notes:

http://math.stanford.edu/~vakil/0708-216/

http://math.stanford.edu/~vakil/0910-216/

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Liu wrote a nice book, which is a bit more oriented to arithmetic geometry. (The last few chapters contain some material which is very pretty but unusual for a basic text, such as reduction of algebraic curves.)

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I'm a fan of The Geometry of Schemes by Eisenbud and Harris. Its great for a conceptual introduction that won't turn people off as fast as Hartshorne. However, it barely even mentions the concept of a module of a scheme, and I believe it ignores sheaf cohomology entirely.

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It does, but it also talks about representability of functors, and does a lot of basic constructions a lot more concretely and in more detail than Hartshorne. – Charles Siegel Oct 25 at 14:53
Oh, I'm a big fan of the book. I'm just warning that if you read it all the way through, you still won't know the 'basics' of algebraic geometry. – Greg Muller Oct 25 at 16:14
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Perhaps this is cliché, but I recommend EGA (links to full texts: I, II, III(1), III(2), IV(1), IV(2), IV(3), IV(4)).

I know it's a scary 1800 pages of French, but

  1. It's really easy French. I would describe myself as not knowing any French, but I can read EGA without too much trouble.
  2. It's extremely clear. The proofs are usually very short because the results are very well organized.
  3. It's the canonical reference for algebraic geometry. I assure you it is not 1800 pages of fluff.

I've found it quite rewarding to to familiarize myself with the contents of EGA. Many algebraic geometry students are able to say with confidence "that's one of the exercises in Hartshorne, chapter II, section 4." It's even more empowering to have that kind of command over a text like EGA, which covers much more material with fewer unnecessary hypotheses and with greater clarity. I've found this combined table of contents to be useful in this quest.

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Has nobody ever considered trying to translate it to English? – fpqc Dec 17 at 3:52
Some time ago I had the idea of starting an EGA translation wiki project. The Berkeley math dept requires its grad students to pass a language exam which consists of translating a page of math in French, German, or Russian into English. I'm sure that many other schools have similar requirements. So every year, we have hundreds of grad students translating a page of math into English. Why not produce something useful with those man-hours? In lieu of a language exam, have the students translate a few pages of EGA. We'd be able to produce a translation of EGA and other works fairly quickly. – Kevin Lin Dec 17 at 12:18
See mathoverflow.net/questions/9171/… – Kevin Lin Dec 17 at 12:36
"The proofs are usually very short because the results are very well organized." This is only one half of the truth!! When I have to look up something in EGA, it's like an infinite tree of theorems which I have to walk up. Every step seems to be trivial, yeah. I don't get the point till I work it out by myself. I'm really envious of the people who learn directly from the master Grothendieck. – Martin Brandenburg Feb 2 at 0:08
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I've been teaching an introductory course in algebraic geometry this semester and I've been looking at many sources. I've found that Milne's online book (jmilne.org) is excellent. He gives quite a thorough treatment of the theory of varieties over an algebraic closed field. The book is very complete and everything seems to be done "in the nicest way".

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vote up 9 vote down

Shafarevich wrote a very basic introduction, it's used in undergraduate classes in algebraic geometry sometimes

Basic Algebraic Geometry 1: Varieties in Projective Space

also, for a more computational point of view

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra

And the followup by the same authors

Using Algebraic Geometry

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The Cox, Little, O'Shea books are what I use when introducing the subject to someone with less background, or more concrete interests. They tend to work very well (advising a freshman through IVA this semester, actually.) – Charles Siegel Oct 25 at 14:25
1 
 Shafarevich also has a Volume 2, on schemes and advanced topics. I'd say that both books are suitable for a graduate-level introduction, and are my vote for best algebraic geometry textbook. – Alison Miller Oct 25 at 20:27
Yes, it might be good idea to include volume 2 in the answer as well, the book is highly readable. – Ilya Nikokoshev Oct 25 at 22:02
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Joe Harris's book Algebraic Geometry might be a good warm-up to Hartshorne.

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I enjoyed Griffiths-Harris a lot.

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At a lower level then Hartshorne is the fantastic "Algebraic Curves" by Fulton. It's available on his website.

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The book An Invitation to Algebraic Geometry by Karen Smith et al. is excellent "for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites," to quote from the product description at amazon.com.

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Also Eisenbud.

Every algebraic geometer needs to know at least some commutative algebra. And this is a very good introductory textbook, which teaches commutative algebra rigorously but at the same time provides a good geometric explanation.

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This isn't really an algebraic geometry textbook. I think it is useful for algebraic geometers, but you should add an explanation of what is useful about it. As is, the only people who can appreciate this answer are the people who already know what you're trying to tell them. – Anton Geraschenko Oct 25 at 18:46
Hope this makes my post more clear. – Ilya Nikokoshev Oct 25 at 19:25
Yes, that's much better. Thanks. – Anton Geraschenko Oct 25 at 21:49
Personally, I prefer Matsumura's "Commutative Ring Theory" to Eisenbud's book. Only problem I have with it, is the slightly annoying layout. – Lars Oct 29 at 22:27
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I believe the issue of "which book is best" is extremely sensitive to the path along which one is moving into the subject. If your background is in differential geometry, complex analysis, etc, then Huybrechts' Complex Geometry is a good bridge between those vantage points and a more algebraic geometric landscape. Obviously I'm taking liberties with the question, as I wouldn't advertise Huybrechts' book as an algebraic geometry text in the strict sense. However, I think it can, for certain people, help to ease the transition into one. It's also very well written, in my opinion. (I should also emphasize that I'm not saying this is the only purpose of the book: its content is extremely valuable for other reasons, with material on vector bundles, SUSY, deformations of complex structures, etc.)

As for dedicated algebraic geometry texts other than Hartshorne, I also vote for Ravi Vakil's notes. They're excellent.

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If Griffiths-Harris is "algebraic geometry" then surely Huybrechts is as well! :) Even if your aim is to learn more abstract scheme theory, I think it's very important and helpful (at least it has been for me) to gain some intuition by learning about complex manifolds and varieties. It also provides some historical context. – Kevin Lin Dec 17 at 11:59
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Computer Scientists, me included, seem to prefer Ideals, Varieties, and Algorithms by David A. Cox, John B. Little, Don O'Shea (http://www.cs.amherst.edu/~dac/iva.html)

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I second Shafarevitch's two volumes on Basic Algebraic Geometry: the best overview of the subject I have ever read.

Another very nice book is Miranda's Algebraic Curves which manages to get a long way (Riemann-Roch etc) without doing sheaves and line bundles until the end. Of course, by then, you are really wanting sheaves and line bundles!

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I've also heard very great things about Miranda's book. It clearly is a less advanced book, but I've heard it makes great preparation for understanding more modern algebraic geometry (e.g. Hartshorne). – Davidac897 Jan 3 at 22:26
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Kenji Ueno's three-volume "Algebraic Geometry" is well-written, clear, and has the perfect mix of text and diagrams. It's undoubtedly a real masterpiece- very user-friendly.

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Also lots of things on jmilne.org

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Macdonald "Algebraic geometry: Introduction to schemes" (not only about noetherian schemes), Dieudonné's two booklets with focus on the motivation and history, the first chapter in Demazure, Gabriel "Groupes algebraique I" (recommended here to undergraduates as intro text), Mumford's "red book".

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If you know french, you might enjoy David Harari's course notes. These are the notes for a basic course in schemes and cohomology of sheaves. He combines the best parts of Hartshorne with the best parts of Liu's book. Hartshorne doesn't always do things in the nicest possible way, and the same is of course true for Liu.

I agree that Vakil's notes are great, since they also contain a lot of motivation, ideas and examples. But does anyone know where to get the files with this year's notes? I only found the notes of previous years on the web.

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4 
 He's not posting them online yet; he's been handing out chunks of notes on various topics, but he wants to edit them more before posting. – Rebecca Bellovin Oct 25 at 21:25
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Miles Reid's Undergraduate Algebraic Geometry is an excellent topical (meaning it does not intend to cover any substantial part of the whole subject) introduction. In particular, it's the only undergraduate textbook that isn't commutative algebra with a few pictures thrown in.

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Mukai's Introduction to Invariants and Moduli surely deserves to be on this list.

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I liked Mumford's "Algebraic geometry I: Complex projective varieties" a lot, and also Griffiths' "Introduction to algebraic curves". Now I think I am falling in love with "Griffiths & Harris". For the record, I hate Hartshorne's.

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