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.

• 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. Oct 25, 2009 at 16:02
• 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. Oct 25, 2009 at 21:52
• 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. Oct 27, 2009 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. Dec 17, 2009 at 3:50
• -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. Jun 1, 2010 at 20:54

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: GĂ¶rtz; 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: KollĂˇr - 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/KovĂˇcs' "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: KollĂˇr; 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.
– Mark
Nov 9, 2011 at 1:03
• 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). Jun 3, 2016 at 20:43
• 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?
– Jam
Nov 7, 2017 at 16:33
• The URL reference to the Gathmann lecture notes appears to be broken. This one looks fine mathematik.uni-kl.de/~gathmann/class/alggeom-2002/… Nov 12, 2018 at 20:00

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

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

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

• 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. Jul 7, 2010 at 5:38
• I think it is important to have links to the newest version: math216.wordpress.com and actual PDFs at math.stanford.edu/~vakil/216blog. Nov 3, 2011 at 7:01
• vakils notes are not motivating, if you are already strugling with definitions without seeing enough motivation and examples and approaches it is not good. Nov 30, 2021 at 2:18

Perhaps this is clichĂ©, 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.]

• 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. Dec 17, 2009 at 12:18
• "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. Feb 2, 2010 at 0:08
• 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. Jun 2, 2010 at 1:04
• @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? Jun 2, 2010 at 16:17
• 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. Jun 3, 2010 at 17:59

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. Oct 25, 2009 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. Oct 25, 2009 at 16:14
• 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? Aug 18, 2010 at 15:53

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. Jul 7, 2010 at 5:56
• I love Part 1 and Part 3 of Liu's book, but I believe that another reference is necessary for cohomology.
– ACL
Jan 7, 2013 at 21:50

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.) Oct 25, 2009 at 14:25
• 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. Oct 25, 2009 at 20:27
• Yes, it might be good idea to include volume 2 in the answer as well, the book is highly readable. Oct 25, 2009 at 22:02
• @ Alison I second your vote,Alison. Jun 1, 2010 at 21:16
• I totally, absolutely agree about Shafarevitch being the best textbook. Sep 29, 2011 at 2:36

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. Jul 7, 2010 at 5:31

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. Jun 1, 2010 at 21:11
• 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. Nov 2, 2011 at 17:46
• ...and if one can read mathematical Japanese, which requires very little working knowledge of the language, the original Japanese text is even more compact (just one volume!) Jun 13, 2020 at 1:57

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 am SHOCKED that this book hasn't gotten more votes, it's very geometric and an easier read than Shafarevich (which I also like very much). Is it a symptom of groupthink or a tendency of each generation to pick their own idols? Jun 2, 2010 at 0:57

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 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). Jan 3, 2010 at 22:26
• Miranda looks very good,although I haven't read it carefully yet. And Shafarevitch right now,to me,is your best bet for serious graduate students. Beautifully written,comprehensive and not too abstract. Jun 1, 2010 at 21:15

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)

• For people with an interest in practical aspects of AG, what about Abhyankar's Algebraic geometry for scientists and engineers? Aug 18, 2010 at 15:57

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.

• I would think Algebraic Geometry 2 would be the successor to Algebraic Geometry 1 and not Red Book. Also any news on when Algebraic Geometry 2 will be published? Feb 5, 2011 at 11:42

Joe Harris's book Algebraic Geometry might be a good warm-up to Hartshorne.

I'm starting to like this book, by GĂ¶rtz and Wedhorn. (and hoping in volume II soon...) Similarly to Qing Liu's wonderful book, it seems to me to be a good compromise between Hartshorne and EGA.

I enjoyed Griffiths-Harris a lot.

• How about the wrong definition of a sheaf which survived all editions of Griffiths-Harris inlcuding Russian translation (they extend from compatible pairs, not arbitrary compatible families of local sections: thus the non-sheaf examples like presheaf of bounded functions would do). Oct 29, 2010 at 21:15

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.

• 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. Dec 17, 2009 at 11:59
• Steven said what I think way better than I can. Jul 7, 2010 at 6:00

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.

• I agree,both of Reid's texts are fantastic introductions.But we don't really have a good,deep text for advanced students yet.Hartshorne-I'm sorry,Professor Hartshorne-is ridiculously abstract and has acted as a torture device for graduate students for far too long. Jun 1, 2010 at 21:10
• The uniqueness claim is a bit strong: what about Mumford, for example? Jun 2, 2010 at 1:00
• Not to mention Qing Liu's book... Jun 2, 2010 at 2:50
• The only differences between the first and second editions of Mumford's Red Book are the numerous typographical errors introduced during its incompetent TeXing... Jun 4, 2010 at 3:36
• Dear Andrew L, Regarding your first comment: when I was a student learning from Hartshorne, I had various complaints about it, but on the other hand, I also learned a vast amount from it. And I've grown more and more to appreciate its very beautiful (and not at all abstract) treatment of curves and surfaces in Chapters 4 and 5. On the other hand, as a student my complaint was that it was not abstract enough (didn't treat non-alg. closed fields, finite flat group schemes over integer rings, abelian schemes, flat descent, etc.). Jul 9, 2010 at 2:12

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.

• 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. Oct 25, 2009 at 18:46
• Hope this makes my post more clear. Oct 25, 2009 at 19:25
• 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, 2009 at 22:27
• Eisenbud's book is wonderfully written and a pleasure to read,but it's too damn long and has everything in the world in it,making it really tough to focus with. It joins Spivak and Lee's SMOOTH MANIFOLDS with the dubious distinction of being books everyone loves,but can't really use for coursework. Jul 7, 2010 at 5:41
• @Andrew L wait, why can't people use Lee?! Mar 2, 2011 at 3:03

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

• The link to the PDF isn't working for me. Jul 2, 2015 at 5:31
• I moved it to five-dimensions.org/Textbooks Jul 3, 2015 at 13:46

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!

Also lots of things on jmilne.org

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.

• 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. Oct 25, 2009 at 21:25

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!

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.

• Excellent book indeed! It is a pleasure to read as an introduction to commutative algebra, algebraic number theory and algebraic geometry through the unifying theme of arithmetic. One of my favorites. Oct 17, 2013 at 21:24

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).

Macdonald "Algebraic geometry: Introduction to schemes" (not only about noetherian schemes), DieudonnĂ©'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 Macdonald book is really good. Aug 6, 2010 at 22:48

Hodge, Pedoe, Methods of Algebraic Geometry.

• Out of curiosity, can you elaborate on what you like about this book? I must admit that I find it almost unreadable, owing to the old-fashioned language.
– user5117
Jul 15, 2013 at 18:35
• Artie, that's exactly what I like about it. I am an old person, and find Hartshorne almost unreadable. Jul 15, 2013 at 21:14
• Fair enough! I wish I could understand it better --- there are interesting things there that I can't find elsewhere. I guess I need to learn the language of primals and object-varieties and Cayley forms...
– user5117
Jul 15, 2013 at 21:32

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