The original motivation of the author was to give an exposition of arithmetic surfaces. This is in fact two rather separate books which have been reprinted in a single volume. It is still a very good introduction, written in the author's characteristic style: informality paired with precision. A substantial text of about pages. The book has detailed proofs, often accompanied by enlightening discussions.

Lectures in Mathematics. And for the brave:. This is the fundamental source. Only 4 chapters of the planned 13 have appeared, but they already comprise about pages. Go for it if you want rigor and generality it has been translated into Chinese! This Wiki based enterprise is becoming the natural successor of EGA as the standard opus of reference for algebraic geometry.

It is certainly as rigorous and general and goes well beyond the notion of a scheme. These course notes delve into the subject in a true Grothendieck spirit right from the start, yet do this in a way that makes prerequisites minimal. So be careful with your choice of words. The deadline is Friday, Jan. Homework Each week I give homework in the form of some exercises. Handing in homework is not obligatory see below.

If you get grade W for the written exam, then your final grade is computed according to the rule 0. Material covered and exercises starred exercises will be graded and count towards H Sept. 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.

## Introduction to Algebraic Geometry

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. I recently completed a book on algebraic geometry. It is also available in paperback: Amazon listing. 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 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.

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! 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. Aimed at beginning graduate students, it treats Number Fields and Algebraic curves simultaneously. Read Math Overflow. Whereas it is actually not quite a textbook, it is becoming a very popular reference.

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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.

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Best Algebraic Geometry text book? Asked 9 years, 11 months ago. Active 3 months ago. Viewed k times. 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.

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. 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? 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. 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 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!

I also like how he often compares the theorems and definitions with the analogues ones theorems or definitions in differential or complex geometry. Jun 3 '16 at This one looks fine mathematik. Kevin H. 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.

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I know it's a scary pages of French, but It's really easy French. It's extremely clear. The proofs are usually very short because the results are very well organized. It's the canonical reference for algebraic geometry. I assure you it is not pages of fluff. 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.

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We'd be able to produce a translation of EGA and other works fairly quickly. Lin Dec 17 '09 at 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. EGA isn't any more textbook of algebraic geometry than Bourbaki is a textbook of mathematics.

## Introduction to Algebraic Geometry

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?

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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. Greg Muller. I'm just warning that if you read it all the way through, you still won't know the 'basics' of algebraic geometry. I just wish they kept the original title, Why Schemes? Michael Hoffman. They tend to work very well advising a freshman through IVA this semester, actually.

I'd say that both books are suitable for a graduate-level introduction, and are my vote for best algebraic geometry textbook. Daniel Moskovich. At least, I may get some basic notions fastly and also see some concrete examples. Joel Kamnitzer. Fran Burstall. It clearly is a less advanced book, but I've heard it makes great preparation for understanding more modern algebraic geometry e. And Shafarevitch right now,to me,is your best bet for serious graduate students. Beautifully written,comprehensive and not too abstract. Is it a symptom of groupthink or a tendency of each generation to pick their own idols?

John D. Jakub Marecek. Also any news on when Algebraic Geometry 2 will be published? Ilya Nikokoshev. Alexander Woo. 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. 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. 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. Only problem I have with it, is the slightly annoying layout. Bischof Mar 2 '11 at It also provides some historical context. Marco Garuti. 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. Alexandre Eremenko.