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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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Since from my point of view the subject is too broad to choose only one book, I posted an answer with my choices for "best book" depending on the focus one is approaching algebraic geometry... Hope that is ok, if not, I shall remove it. –  Javier Álvarez Mar 1 '11 at 18:31
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10 Answers

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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Ravi Vakil will be editing and posting his notes over the coming year; see math216.wordpress.com. –  Moor Xu Jul 29 '10 at 2:54
" Also there are tons of exercises" this is why I love these notes! +1 –  BBischof Jul 30 '10 at 4:16
Hey 'blo. I personally think EGA is much easier than Hartshorne! EGA is like the Hartshorne solutions manual, basically. ;-) –  Kevin H. Lin Jul 31 '10 at 0:34
@Kevin when I said 'baby version of EGA' I mean the scope and generality of EGA is more extensive, not more difficult to understand. Some people say EGA is reason enough to learn French. But I think the books and notes mentioned here show that you don't have to learn French. –  solbap Jul 31 '10 at 20:06
Vakil's notes look like a remarkable work and when it reaches something like a complete form-according to the author,it may remain online in perpetuity,so it may never really reach a "published"form in the conventional sense-it may very well replace Hartshorne as the standard source on the subject. –  Mathemagician1234 Feb 15 '12 at 22:19
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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" 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 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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One suggestion per answer? –  Bruno Stonek Mar 1 '11 at 18:14
I am sorry and will remove the answer if the poster wants it so. I was just trying to be complete in the sense that the best book on algebraic geometry besides Hartshorne is not only one, but depends on the level or subject within Algebraic Geometry you are referring to. For example, Hartshorne's is not at all the best book for some physicists doing string theory, so in that case Griffiths/Harris suits best. Instead of posting 13 answers reflecting these distinctions, I just posted all inside one answer. I hope it is ok?.... if not, excuse me again. –  Javier Álvarez Mar 1 '11 at 18:27
@Javier: no problem with more than on suggestion. I found it very informative. –  Marc Olschok Feb 17 '12 at 16:02
@JavierÁlvarez I'm in a similar situation as OP. But I have a more specific question, how would you compare Hartshorne's book Algebraic Geometry to Algebraic Geometry I, Schemes with Examples by Gortz and Wedhorn? I have never had a course in Algebraic Geometry before, but I have had a graduate course in commutative algebra(Atiyah MacDonald). I was thinking of starting with Chapter I of Hartshorne, and then move on to Gortz book on Sheaves and Schemes...would this be a good approach in your opinion? –  V-B Feb 1 at 20:21
@V-B well, Görtz & Wedhorn will be the best reference to learn 'just' sheaves and schemes, and the best reference for all that plus cohomology when someday the volume 2 gets published. Then, both vol. will serve as a more detailed and motivated treatment than Hartshorne's chapters II and III. But as long as there is no vol. II, you can only learn about schemes in Görtz & Wedhorn, whereas a deep understanding of varieties, sheaf cohomology, curves and surface is needed if you want an equivalent course to Hartshorne's. I recommend you take a look at math.stackexchange.com/a/257528/4058 –  Javier Álvarez Feb 2 at 10:05
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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.

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Nobody has mentioned the book of Shafarevich; so I mention myself.

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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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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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Just started reading bits of this online last night, its very readable (not the scan though). –  Jonathan Fischoff Jul 29 '10 at 2:17
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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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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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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.

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I also specifically recommend this for the profusion of examples...in the course Joe Harris teaches from which these notes initially grew, he explicitly says that his goal for this first course is to give a ton of examples that need look at only later in terms of the modern scheme-theoretic interpretation. –  Jamie Banks Jul 29 '10 at 2:13
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