### Best algebraic geometry textbook? (other than Hartshorne)

• 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.

Community wiki this?

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.

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.

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.

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.

I don't think that Hartshorne is the best book on AG, because he always assumes the schemes to be (locally) noetherian. I know that most algebraic geometers are only interested in these schemes, but this assumption partly overshadows the core of some concepts. Also, very important concepts are dispelt to the exercises ...

-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.

For my own purposes, I like - Dino Lorenzini's book *An Invitation to Arithmetic Geometry" - Q. Liu's "Algebraic Geometry and Arithmetic Curves", Eisenbud's "Geometry of Schemes" - EGA / SGA / FGA 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.

For those who can read Russian, I can't recommend Dmitry Kaledin's course (http://www.mi.ras.ru/~kaledin/noc/) enough. The exposition is very high-level, emphasizing the concepts, not technicalities; the obsession with the latter, unfortunately, spoils many introductory AG texts.

It looks like at some point people just started ignoring the directive to offer a single resource per answer. This seems to make votes for answers somewhat less informative. Because this question is community Wiki, people should feel free to remove redundant parts of other people's answers.

@Martin: worse than that, smoothness is only defined for schemes of finite type over a *field*! Anyway, I believe that students in AG would suffer more without Hartshorne's book than with it.

for an unmotivated student in AG(like me!) I can tell it seems like playing with some abstruse concepts with no motivation whatsover like separated morphisms blabla! bunches of definitions and the exercises seems to interconnect the definitions.

• 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.

• 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 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, a freely available online course is available by the nicely done new long notes by R. Vakil.

• 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.

Gathmann's lecture notes are indeed great. I had a certain phobia with algebraic geometry for a long time, and the the introduction chapter in his notes is the only thing which made me realize that there was nothing to be scared of. His emphasis on the geometric picture (sometimes literally - there are lots of pictures!) rather than on the algebraic language really made me love algebraic geometry. I also like how he often compares the theorems and definitions with the analogues ones theorems or definitions in differential or complex geometry.

The third edition (2013) of Shafarevich's "Basic Algebraic Geometry" now includes the proof of the Riemann-Roch theorem for curves (volume 1, chapter 3, sub-chapter 7).

CLASSICAL: Beltrametti et al. "Lectures on Curves, Surfaces and Projective Varieties" this book has as as prerequisite projective geometry any suggestion what to read before going to the book itself?

The URL reference to the Gathmann lecture notes appears to be broken. This one looks fine https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002.pdf

• Professor Vakil has informed people at his site that this year's version of the notes will be posted in September at his blog.I think these notes are quickly becoming legendary,like Mumford's notes were before publication. A super,2 year long graduate course using totally free materials could begin with Fulton and then move on to Vakil's notes.

I think it is important to have links to the newest version: http://math216.wordpress.com/ and actual PDFs at http://math.stanford.edu/~vakil/216blog/.

• 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. [Edit: The combined table of contents unfortunately seems to be defunct. Here is a web version of Mark Haiman's EGA contents handout.]

Has nobody ever considered trying to translate it to English?

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.

"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.

Excuse me Anton, but you have very perverse sense of what constitutes a textbook. EGA isn't any more textbook of algebraic geometry than Bourbaki is a textbook of mathematics.

@Martin: It's tough for those of us to are trying to "pick up" EGA after learning extensively from some other text. *But* it has the huge advantage that it generalizes well. Indeed, much progress in algebraic geometry has been made by identifying a new kind of geometric object (say algebraic spaces) and "redoing EGA" for them. That is, proving a few linchpin results about them and following the cascade of deductions in EGA to end up with a vast theory.

@Victor: I don't understand your objection. Could you explain in what ways EGA does not constitute a textbook? You certainly don't need to already know algebraic geometry to read it. Reading it, you will certainly learn algebraic geometry. Is your objection that there aren't any exercises? Is it that EGA also covers a lot of commutative algebra, which you'd rather think of as a separate subject? Is it the length? Why is it any worse than Eisenbud's 800 page commutative algebra book plus Griffiths & Harris' 900 page algebraic geometry book?

It's a research monograph (and it's unfinished, by the way). It does build the subject from the ground up, just like Bourbaki's "Elements of mathematics" builds mathematics from the ground up, but it is less pedagogical by comparison (which is understandable). The fact that there are no exercises in it and the manner in which it was written are probably reflections of its function. Note that I don't object that it's a good reference on the foundations of algebraic geometry; but to call it a $\textit{textbook}$, and even nominate it as a *best* AG textbook, is simply preposterous.

For an Introduction to Algebraic Geometry you can also see this book: http://www.ams.org/bookstore-getitem/item=HIN-7

@Tyler: yes, I noticed that too ... now I wish I'd made a copy somewhere :-(. I've turned Mark Haiman's EGA contents handout into a web page; I'll edit that in.

@VictorProtsak When someone tells me, complaining, "there's no exercises", I always answer : there are theorems. Having read basic definitions, you could start proving yourself, propositions, lemmas etc. That's also an exercise, and a very good one. Specializing this argument to EGA's (and also SGA's) you will note that many exercises of all books about algebraic geometry are often extracted from EGA's. By the way, no one complains about the fact that there are no exercises in Berkovich's books, but people nevertheless learn and undestand its theory.

• 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.)

I actually love Liu's approach.

I love Part 1 and Part 3 of Liu's book, but I believe that another reference is necessary for cohomology.

• 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.

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.

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.

Too few textbooks motivate mathematical machinery (not just in AG), so this book really stands apart for that reason. I just wish they kept the original title, *Why Schemes?*

• 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

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.)

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.

Yes, it might be good idea to include volume 2 in the answer as well, the book is highly readable.

@ Alison I second your vote,Alison.

I totally, absolutely agree about Shafarevitch being the best textbook.

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

This is a terrific book from what I've read of it and it will be my first choice when I start seriously relearning this material.

• 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.

I haven't seen it yet,but I've heard a lot of nice things about it from some friends at Oxford,where apparently it's quite popular.

Yes, I think it is quite well-written and easy to proceed . . . and very thin. At least, I may get some basic notions fastly and also see some concrete examples.

• 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".

• 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.