Lifted from Mathoverflow:
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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Before Hartshorne's book there was Mumford's Red Book of Varieties. I think it is a great introductory textbook to modern algebraic geometry (scheme theory).
I found that Mumford is quite good at motivating new concepts; in particular I really enjoy his development of nonsingularity and the sheaf of differentials. I think another great aspect about this book is that it emphasizes how to define things intrinsically (i.e. without reference to a closed or open immersion into affine space) but also explains how to make local arguments (i.e. using immersion into affine space). A classic example of the above:
(non intrinsic tangent space): Say X is a variety and p is a point of X. Choose an affine neighborhood so that p corresponds to the origin. Then this affine neighborhood is spec k[x1, ..., xn]/I for some ideal. Let I' be all the linear terms of I (i.e. if I = (x,y^2), then I' = (x)). Then the tangent space at p is spec k[x1,...,xn]/I'.
(intrinsic tangent space): Let m be the maximal ideal of the local ring of the structure sheaf at p, then the tangent space is the dual of the vector space m/m^2.
Taking spec of the symmetric algebra of the latter gives you the former.
Some drawbacks. This book doesn't cover nearly as much as Hartshorne's book. It doesn't have that many exercises. The notation is slightly different; integral finite type schemes are called pre-varieties and you can remove the `pre' if it's also separated. Nevertheless I think its a great compliment to reading Hartshorne.
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Another book I wish I had known about when I was first reading Hartshorne is Miranda's Complex Algebraic Curves.
Again this book covers much less then Hartshorne and only discusses curves over the complex numbers (and their Jacobians). But it gives a lot more details and examples of concepts which I found particularly difficult when I first started learning algebraic geometry (sheafs, divisors, cohomology). It also has a bunch of exercises which I think are often not as challenging as the the exercises in Hartshorne.
It also covers a lot more of the 'classical' theory of curves than Hartshrone does; e.g. Weierstrass points.
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My last suggestion would be Ravi Vakil's online notes on the foundations of algebraic geometry.
I think these notes might be made into a full on textbook someday. I haven't looked through all of them but these notes seem to cover as much as Hartshorne does (if not more). Only rarely do Hartshorne and Vakil define things differently (`projective morphisms' is the only example that comes to mind).
I've heard it said that Hartshorne's book is a `baby' version of EGA. I think Vakil's notes are somewhere between Hartshorne and EGA (probably not the midpoint though). At least Vakil discusses much more the theory of representable functors, and Noetherian hypothesis are less prevalent in Vakil's notes. Also Vakil's notes are more complete in that they also include proofs of many of the commutative algebra results that are just stated in Hartshorne.
I think Vakil spends a lot more time motivating the material and often the notes are a bit conversational. Also there are tons of exercises and most of the them are appended with useful qualifiers like (easy but important exercises, unimportant exercise, tedious but useful exercise, etc).
One drawback is that they are very long and they are online notes so there are many typos. But most of them are grammatical and easy to spot. [Edit: By now there are only a few typos (because these are online notes)]
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for Undergraduate algebraic geometry (significantly below the level of Hartshorne), Cox, Little and O'Shea's Ideals, Varieties, and Algorithms is a pleasant treatment.
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I'm really enjoying Andreas Gathmann's lecture notes. They are pretty elementary and surprisingly complete (for lecture notes).
Reid also has a really nice text on algebraic geometry («Undergraduate algebraic geometry»).
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I think Algebraic Geometry is too broad a subject to choose only one book. But my personal choices for the BEST BOOKS are
UNDERGRADUATE: Beltrametti et al. "Lectures on Curves, Surfaces and Projective Varieties" (errata) 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.
HALF-WAY: Shafarevich - "Basic Algebraic Geometry" vol. 1 and 2. They are the most complete on foundations and introductory into Schemes so they are very useful before more abstract studies. But the problems are almost impossible.
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. Filled with exercises.
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.
ONLINE NOTES: Gathmann - "Algebraic Geometry" which can be found here. Just amazing notes; short but very complete, dealing even with schemes and cohomology and proving Riemann-Roch. It is the best free book you need to get enough algebraic geometry to understand the other titles.
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. It does a great job complementing Hartshorne's treatment of schemes, above all because of the more solvable exercises. A second volume is on its way on cohomology.
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 notes, it is by far the shortest and manageable 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.
GRADUATE ON ALGEBRAIC CURVES: Arbarello; Cornalba; Griffiths; Harris - "Geometry of Algebraic Curves" vol 1 and vol 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 already needed is minimum compared to other titles.
ADVANCED ON ALGEBRAIC SURFACES: Badescu - "Algebraic Surfaces". For those needing a companion and expansion to Hartshorne's chapter. Done with more advanced tools than Beauville.
ON INTERSECTION THEORY: Fulton - Intersection Theory. By far the best and most complete book on the subject, from general Bézout's theorem to Grothendieck-Riemann-Roch theorem. Lots of examples.
ON RESOLUTION OF SINGULARITIES: Kollár - Lectures on Resolution of Singularities. Small but fundamental book on singularities, methods of resolution for curves and surfaces and proof of the cornerstone Hironaka's theorem. The only main alternative is Cutkosky's book.
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. oriented for complex geometry or for physicists) than what a student of AG from Hartshorne's book may like to learn the subject. The alternative and easier title is Sernesi's book on deformation of algebraic schemes
ON GEOMETRIC INVARIANT THEORY: Mumford; Fogarty; Kirwan - "Geometric Invariant Theory". Simply put, it is the original reference. Besides, Mumford himself developed the subject. The best alternative to this and the previous title, but more on the introductory side, is Mukai's book on moduli and invariants.
ON HIGHER-DIMENSIONAL VARIETIES: Debarre - "Higher Dimensional Algebraic Geometry". The main alternative to this title is Kollar/Mori "Birational Geometry of Algebraic Varieties" but is regarded as much harder to understand by many students. This is a very active frontier of research, with new fundamental results proved in Hacon/Kovács' "Classification of Higher-Dimensional Varieties".
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I possibly cannot say what is "the best" book in this topic, but I've recently started studying it and found Hartshorne's book extremely difficult, so I went to study Mumford's red book of varieties. But other than these books that have been introduced I found the followings also helpful:
A Royal Road to Algebraic Geometry by Audun Holme is a newly published book which tries to make Algebraic Geometry as easy as possible for studetns.
Also, the book by Griffits and Harris called Principles of Algebraic Geometry in spite of being rather old, and working mostly with only complex field, gives a good intuition on this very abstract topic.
Algebraic Geometry: A First Course by Joe Harris is a very good book that sits in that region between undergraduate treatments and the prerequisites of Hartshorne. In particular, one does not need to know much commutative algebra to get a lot out of Harris's book. Harris himself recommends reading Hartshorne after his book for the theory of schemes.