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.
I think Algebraic Geometry is too broad a subject to choose only one book. Maybe if one is a beginner then a clear introductory book is enough or if algebraic geometry is not ones major field of study then a self-contained reference dealing with the important topics thoroughly is enough. But Algebraic Geometry nowadays has grown into such a deep and ample field of study that a graduate student has to focus heavily on one or two topics whereas at the same time must be able to use the fundamental results of other close subfields. Therefore I find the attempt to reduce his/her study to just one book (besides Hartshorne's) too hard and unpractical. That is why I have collected what in my humble opinion are the best books for each stage and topic of study, my personal choices for the best books are then:
CLASSICAL: Beltrametti-Carletti-Gallarati-Monti. "Lectures on Curves, Surfaces and Projective Varieties" which starts from the very beginning with a classical geometric style. Very complete (proves Riemann-Roch for curves in an easy language) and concrete in classic constructions needed to understand the reasons about why things are done the way they are in advanced purely algebraic books. There are very few books like this and they should be a must to start learning the subject. (Check out Dolgachev's review.)
HALF-WAY/UNDERGRADUATE: Shafarevich - "Basic Algebraic Geometry" vol. 1 and 2. They may be the most complete on foundations for varieties up to introducing schemes and complex geometry, so they are very useful before more abstract studies. But the problems are hard for many beginners. They do not prove Riemann-Roch (which is done classically without cohomology in the previous recommendation) so a modern more orthodox course would be Perrin's "Algebraic Geometry, An Introduction", which in fact introduce cohomology and prove RR.
ADVANCED UNDERGRADUATE: Holme - "A Royal Road to Algebraic Geometry". This new title is wonderful: it starts by introducing algebraic affine and projective curves and varieties and builds the theory up in the first half of the book as the perfect introduction to Hartshorne's chapter I. The second half then jumps into a categorical introduction to schemes, bits of cohomology and even glimpses of intersection theory.
FREE ONLINE NOTES:
Gathmann - "Algebraic Geometry" (All versions are found here. The latest is of 2019.) Just amazing notes; short but very complete, dealing even with schemes and cohomology and proving Riemann-Roch and even hinting Hirzebruch-R-R. It is the best free course in my opinion, to get enough algebraic geometry background to understand the other more advanced and abstract titles.
For an abstract algebraic approach, the nice, long notes by Ravi Vakil is found here. (A link to all versions; the latest is of 2017.)
GRADUATE FOR ALGEBRISTS AND NUMBER THEORISTS: Liu Qing - "Algebraic Geometry and Arithmetic Curves". It is a very complete book even introducing some needed commutative algebra and preparing the reader to learn arithmetic geometry like Mordell's conjecture, Faltings' or even Fermat-Wiles Theorem.
GRADUATE FOR GEOMETERS: Griffiths; Harris - "Principles of Algebraic Geometry". By far the best for a complex-geometry-oriented mind. Also useful coming from studies on several complex variables or differential geometry. It develops a lot of algebraic geometry without so much advanced commutative and homological algebra as the modern books tend to emphasize.
BEST ON SCHEMES: Görtz; Wedhorn - Algebraic Geometry I, Schemes with Examples and Exercises. Tons of stuff on schemes; more complete than Mumford's Red Book (For an online free alternative check Mumfords' Algebraic Geometry II unpublished notes on schemes.). It does a great job complementing Hartshorne's treatment of schemes, above all because of the more solvable exercises.
UNDERGRADUATE ON ALGEBRAIC CURVES: Fulton - "Algebraic Curves, an Introduction to Algebraic Geometry" which can be found here. It is a classic and although the flavor is clearly of typed concise notes, it is by far the shortest but thorough book on curves, which serves as a very nice introduction to the whole subject. It does everything that is needed to prove Riemann-Roch for curves and introduces many concepts useful to motivate more advanced courses.
GRADUATE ON ALGEBRAIC CURVES: Arbarello; Cornalba; Griffiths; Harris - "Geometry of Algebraic Curves" vol 1 and 2. This one is focused on the reader, therefore many results are stated to be worked out. So some people find it the best way to really master the subject. Besides, the vol. 2 has finally appeared making the two huge volumes a complete reference on the subject.
INTRODUCTORY ON ALGEBRAIC SURFACES: Beauville - "Complex Algebraic Surfaces". I have not found a quicker and simpler way to learn and clasify algebraic surfaces. The background needed is minimum compared to other titles.
ADVANCED ON ALGEBRAIC SURFACES: Badescu - "Algebraic Surfaces". Excellent complete and advanced reference for surfaces. Very well done and indispensable for those needing a companion, but above all an expansion, to Hartshorne's chapter.
ON HODGE THEORY AND TOPOLOGY: Voisin - Hodge Theory and Complex Algebraic Geometry vols. I and II. The first volume can serve almost as an introduction to complex geometry and the second to its topology. They are becoming more and more the standard reference on these topics, fitting nicely between abstract algebraic geometry and complex differential geometry.
INTRODUCTORY ON MODULI AND INVARIANTS: Mukai - An Introduction to Invariants and Moduli. Excellent but extremely expensive hardcover book. When a cheaper paperback edition is released by Cambridge Press any serious student of algebraic geometry should own a copy since, again, it is one of those titles that help motivate and give conceptual insights needed to make any sense of abstract monographs like the next ones.
ON MODULI SPACES AND DEFORMATIONS: Hartshorne - "Deformation Theory". Just the perfect complement to Hartshorne's main book, since it did not deal with these matters, and other books approach the subject from a different point of view (e.g. geared to complex geometry or to physicists) than what a student of AG from Hartshorne's book may like to learn the subject.
ON GEOMETRIC INVARIANT THEORY: Mumford; Fogarty; Kirwan - "Geometric Invariant Theory". Simply put, it is still the best and most complete. Besides, Mumford himself developed the subject. Alternatives are more introductory lectures by Dolgachev.
ON INTERSECTION THEORY: Fulton - "Intersection Theory". It is the standard reference and is also cheap compared to others. It deals with all the material needed on intersections for a serious student going beyond Hartshorne's appendix; it is a good reference for the use of the language of characteristic classes in algebraic geometry, proving Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch among many interesting results.
ON SINGULARITIES: Kollár - Lectures on Resolution of Singularities. Great exposition, useful contents and examples on topics one has to deal with sooner or later. As a fundamental complement check Hauser's wonderful paper on the Hironaka theorem.
ON POSITIVITY: Lazarsfeld - Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series and Positivity in Algebraic Geometry II: Positivity for Vector Bundles and Multiplier Ideals. Amazingly well written and unique on the topic, summarizing and bringing together lots of information, results, and many many examples.
INTRODUCTORY ON HIGHER-DIMENSIONAL VARIETIES: Debarre - "Higher Dimensional Algebraic Geometry". The main alternative to this title is the new book by Hacon/Kovács' "Classifiaction of Higher-dimensional Algebraic Varieties" which includes recent results on the classification problem and is intended as a graduate topics course.
ADVANCED ON HIGHER-DIMENSIONAL VARIETIES: Kollár; Mori - Birational Geometry of Algebraic Varieties. Considered as harder to learn from by some students, it has become the standard reference on birational geometry.
I think the best "textbook" is Ravi Vakil's notes:
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
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. [Edit: The combined table of contents unfortunately seems to be defunct. Here is a web version of Mark Haiman's EGA contents handout.]
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.
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.)
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
And the followup by the same authors
At a lower level then Hartshorne is the fantastic "Algebraic Curves" by Fulton. It's available on his website.
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.
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.
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".
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.
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!
I've tried learning algebraic geometry several times. I asked around and was told to read Hartshorne. I started reading it several times and each time put it away. I realized that I could work through the sections and solve some of the problems, but I gained absolutely no intuition for reading Hartshorne. Discussing this with other people, I found that it was a common occurrence for students to read Hartshorne and afterwards have no idea how to do algebraic geometry. (I imagine this was the motivation for asking this question.)
After more poking around, I discovered Mumford's "Red book of Varieties and Schemes". While Mumford doesn't do cohomology, he motivates the definitions of schemes and and many of there basic properties while providing the reader with geometric intuition. This book isn't easy to read and you have to work out a lot, but the rewards are great. Another great feature of this book is that Mumford bought the rights to the book back from Springer and the book is available for free online.
Another book was supposed to be written that built on the "Red book" including cohomology. After many years, I think this is near completion; see Algebraic Geometry 2. Whlile many of the above books are excellent, it's a surprise that these books aren't the standard.
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)
I enjoyed Griffiths-Harris a lot.
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.
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 H. Lin Dec 17 '09 at 11:59Miles 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.
I recently completed a book on algebraic geometry. The PDF file may be freely downloaded: Introduction to Algebraic Geometry
It is also available in paperback:Amazon listing
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.
Next to the classics, there is Introduction to algebraic varieties and Introduction to Schemes (available here) by Ellingsrud-Ottem. They are the lecture notes, respectively, for the courses Algebraic Geometry I and II taught at the University of Oslo.
Introduction to algebraic varieties is suited for a person that hasn't seen any algebraic geometry before. The text, by means of the introduction of sheaf theory in the more intuitive context of classical (pre)varieties, prepares well the student to the subsequent study of schemes.
On the other hand, the best part about Introduction to Schemes is that many insightful examples are explicitly computed accompanied with nice, colorful pictures. E.g. already chapter 5 (just after introducing schemes!) spans 18 pages of examples about gluing (from $\mathbb{P}^n$ to hyperelliptic curves to Hirzebruch surfaces $\mathbb{F}_r$). Prof. Ottem himself has mentioned that writing out these examples are important to him, and it really shows!
A friendly introduction to Arithmetic Algebraic Geometry is Lorenzini's book "An invitation to Arithmetic Geometry". Aimed at beginning graduate students, it treats Number Fields and Algebraic curves simultaneously.
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.
Biased by my personal taste maybe, I think, Harder's two-volume book(with the third one not completed yet) Lectures on Algebraic Geometry is wonderful. The author develops the algebraic side of our subject carefully and always strikes a good balance between abstract and concrete. If you can torlerate the English written by a German, perhaps some parts of Harder's are more appealing than those of Shafarevich and Hartshorne!
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", Mumford's "red book".
Mumford suggested in a letter to Grothendieck to publish a suitable edited selection of letters from Grothendieck to his friends, because the letters he received from him were "by far the most important things which explained your ideas and insights ... vivid and unencumbered by the customary style of formal french publications ... express(ing) succintly the essential ideas and motivations and often giv(ing) quite complete ideas about how to overcome the main technical problems ... a clear alternative (to the existing texts) for students who wish to gain access rapidly to the core of your ideas". (Found in the very beautifull 2nd collection - when I got it from the library I could not stop reading in it, which happens to me rarely with such collections, despite the associated saga)
The red book by Mumford is nice, better than Hartshorne in my opinion (which is nice as well). At a far more abstract level, EGA's are excellent, proofs are well detailed but intuition is completly absent. For a down to earth introduction, Milne's notes are nice (but they don't go to the scheme level, they give the taste of it).
Hodge, Pedoe, Methods of Algebraic Geometry.
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.
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).
Manin's lectures on algebraic geometry that were recently translated into English could be helpful too.
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...
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)
Another nice introductory AG book that, I believe, was not mentioned here yet is Hassett.
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.
The first book is elementary, down to earth and full of examples.
For scheme theory, have a look at Grothendieck's original Bourbaki papers too. Grothendieck's technical articles are intended to be references that last for eternity, but his early Bourbaki talks were expository, sketching the future landscape of what would become his EGA project. Those talks are great.
– xuros Jul 07 '10 at 08:20