## A learning roadmap for algebraic geometry

Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be useful for other people too. I'm interested in learning modern Grothendieck-style algebraic geometry in depth. I have some familiarity with classical varieties, schemes, and sheaf cohomology (via Hartshorne and a fair portion of EGA I) but would like to get into some of the fancy modern things like stacks, étale cohomology, intersection theory, moduli spaces, etc. However, there is a vast amount of material to understand before one gets there, and there seems to be a big jump between each pair of sources. Bourbaki apparently didn't get anywhere near algebraic geometry.

So, does anyone have any suggestions on how to tackle such a broad subject, references to read (including motivation, preferably!), or advice on which order the material should ultimately be learned--including the prerequisites? Is there ultimately an "algebraic geometry sucks" phase for every aspiring algebraic geometer, as Harrison suggested on these forums for pure algebra, that only (enormous) persistence can overcome?

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Akhil, to be fair to everyone else ever, I have a "foo sucks" phase for pretty much everything: I learn some shiny new facts about something, get heavily into it for a while without understanding it on more than a surface level, and then when I start to learn it "for real," there's more ugly technical details and fewer of the broad, striking generalities that attracted me to the field. So this is more a byproduct of my way of learning things than an absolute necessity. – Harrison Brown Oct 19 2009 at 23:03
Same here, incidentally. But I think the problem might be worse for algebraic geometry---after all, the "barriers to entry" (i.e. theoretical prerequisite material) are somewhat more voluminous than for analysis, no? – Akhil Mathew Oct 20 2009 at 1:04
Maybe interesting: Oort's talk on Grothendiecks mindset: staff.science.uu.nl/~oort0109/… – Thomas Riepe Nov 6 2011 at 12:41

FGA Explained. Articles by a bunch of people, most of them free online. You have Vistoli explaining what a Stack is, with Descent Theory, Nitsure constructing the Hilbert and Quot schemes, with interesting special cases examined by Fantechi and Goettsche, Illusie doing formal geometry and Kleiman talking about the Picard scheme.

For intersection theory, I second Fulton's book.

And for more on the Hilbert scheme (and Chow varieties, for that matter) I rather like the first chapter of Kollar's "Rational Curves on Algebraic Varieties", though he references a couple of theorems in Mumfords "Curves on Surfaces" to do the construction.

And on the "algebraic geometry sucks" part, I never hit it, but then I've been just grabbing things piecemeal for awhile and not worrying too much about getting a proper, thorough grounding in any bit of technical stuff until I really need it, and when I do anything, I always just fall back to focus on varieties over C to make sure I know what's going on.

EDIT: Forgot to mention, Gelfand, Kapranov, Zelevinsky "Discriminants, resultants and multidimensional determinants" covers a lot of ground, fairly concretely, including Chow varieties and some toric stuff, if I recall right (don't have it in front of me)

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Thanks! So you're advising emphatically not to go the EGA-route (i.e. do nothing other than reading Grothendieck linearly for several months, but rather skip arond from different sources)? Perhaps this is the antidote for that phase. – Akhil Mathew Oct 19 2009 at 23:00
I've actually never cracked EGA open except to look up references. SGA, too, though that's more on my list. I'll probably have to eventually, but I at least have a feel for what's going on without having done so, and other people have written good high-level expositions of most of the stuff that Grothendieck did. And specifically, FGA Explained has become one of my favorite references for anything resembling moduli spaces or deformations. – Charles Siegel Oct 19 2009 at 23:03
Gelfand, Kapranov, and Zelevinsky is a book that I've always wished I could read and understand. Maybe this is a "royal road" type question, but what're some good references for a beginner to get up to that level? – jc Oct 21 2009 at 23:47
Well, to get a handle on discriminants, resultants and multidimensional determinants themselves, I can't recommend the two books by Cox, Little and O'Shea enough. The first one, Ideals, Varieties and Algorithms, is undergrad, and talks about discriminants and resultants very classically in elimination theory. The second, Using Algebraic Geometry, talks about multidimensional determinants. – Charles Siegel Oct 22 2009 at 1:11

Concentrated reading on any given topic—especially one in algebraic geometry, where there is so much technique—is nearly impossible, at least for people with my impatient idiosyncracy. It's much easier to proceed as follows.

1. Ask an expert to explain a topic to you, the main ideas, that is, and the main theorems. Keep diligent notes of the conversations.
2. Try to prove the theorems in your notes or find a toy analogue that exhibits some of the main ideas of the theory and try to prove the main theorems there; you'll fail terribly, most likely.
3. Once you've failed enough, go back to the expert, and ask for a reference.
4. Open the reference at the page of the most important theorem, and start reading.
5. Every time you find a word you don't understand or a theorem you don't know about, look it up and try to understand it, but don't read too much. At this stage, it helps to have a table of contents of FGA explained-EGA-SGA where you can quickly look up unknown words. Keep diligent notes of your progress, and talk to your expert as much as possible. Then go back to step 2.

An example of a topic that lends itself to this kind of independent study is abelian schemes, where some of the main topics are (with references in parentheses):

1. the rigidity lemma (Mumford, Geometric invariant theory, Chapter 6),
2. the theorem of the cube (Raynaud, Faisceaux amples sur les schémas…),
3. construction of the dual abelian scheme (Faltings-Chai, Degeneration of abelian varieties, Chapter 1),
4. questions of projectivity (Raynaud, Faisceaux amples sur les schemas…),
5. Lang-Néron theorem and $K/k$ traces (Brian Conrad's notes).
6. proof that abelian schemes assemble into an algebraic stack (Mumford, Geometric invariant theory, Chapter 7),
7. compactifications of the stack of abelian schemes (Faltings-Chai, Degeneration of abelian varieties; Olsson, Canonical compactifications…; Kato and Usui, Classidying spaces of degenerating polarized Hodge structures.)

You may amuse yourself by working out the first topics above over an arbitrary base. That's enough to keep you at work for a few years!

A brilliant epitome of SGA 3 and Gabriel-Demazure is Sancho de Salas, Grupos algebraicos y teoria de invariantes. It explains the general theory of algebraic groups, and the general representation theory of reductive groups using modern language: schemes, fppf descent, etc., in only 400 quatro-sized pages!

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I like the use of toy analogues. Personally, I don't understand anything until I've proven a toy analogue for finite graphs in one way or another. – Qiaochu Yuan Oct 20 2009 at 2:43

I need to go at once so I'll just put a link here and add some comments later. Or someone else will. The Stacks Project - nearly 1500 pages of algebraic geometry from categories to stacks.

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My advice: spend a lot of time going to seminars (and conferences/workshops, if possible) and reading papers. Talk to people, read blogs, subscribe to the arxiv AG feed, etc. AG is a very large field, so look around and see what's out there in terms of current research. There's a huge variety of stuff. There's a lot of "classical" stuff, and there's also a lot of cool "modern" stuff that relates to physics and to topology (e.g. Gromov-Witten theory, derived algebraic geometry). First find something more specific that you're interested in, and then try to learn the background that's needed.

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I'm only an "algebraic geometry enthusiast", so my advice should probably be taken with a grain of salt. With that said, here are some nice things to read once you've mastered Hartshorne.

1) I'm a big fan of Mumford's "Curves on an algebraic surface" as a "second" book in algebraic geometry.

2) Fulton's "Toric Varieties" is also very nice and readable, and will give access to some nice examples (lots of beginners don't seem to know enough explicit examples to work with).

3) More stuff about algebraic curves. The best book here would be "Geometry of Algebraic Curves" by Arbarello, Cornalba, Griffiths, and Harris. The next step would be to learn something about the moduli space of curves. An inspiring choice here would be "Moduli of Curves" by Harris and Morrison.

4) Intersection Theory. Fulton's book is very nice and readable.

5) Algebraic groups. I'm a big fan of Springer's book here, though it is written in the language of varieties instead of schemes.

EDIT : I forgot to mention Kollar's book on resolutions of singularities. A masterpiece of exposition!

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 As for Fulton's "Toric Varieties" a somewhat more basic intro is in the works from Cox, Little and Schenck, and can be found on Cox's website. Also, in theory (though very conjectural) volume 2 of ACGH Geometry of Algebraic Curves, about moduli spaces and families of curves, is slated to print next year. But they said that last year...though the information on Springer's site is getting more up to date. – Charles Siegel Oct 19 2009 at 23:11 I actually possess a preprint copy of ACGH vol II, and Joe Harris promised me that it would be published soon! – Andy Putman Oct 19 2009 at 23:16 Great! I've been waiting for it for a couple of years now. Springer's been claiming the earliest possible release date and then pushing it back. – Charles Siegel Oct 19 2009 at 23:44 And now I wish I could edit my last comment, to respond to your edit: Kollar's book is great. I learned a lot from it, and haven't even gotten to the general case, curves and surface resolution are rich enough. – Charles Siegel Oct 19 2009 at 23:47 I have owned a prepub copy of ACGH vol.2 since 1979. Of course it has evolved some since then. BY now I believe it is actually (almost) shipping. – roy smith May 12 2011 at 21:04
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For me, I think the key was that much of my learning algebraic geometry was aimed at applying it somewhere else. Starting with a problem you know you are interested in and motivated about works very well. For me it was certain bits of geometric representation theory (which is how I ended up learning etale cohomology in the hopes understanding knots better), but for someone else it could be really wanting to understand Gromov-Witten theory, or geometric Langlands, or applications of cohomology in number theory.

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I found that this article "Stacks for everybody" was a fun read (look at the title!), and provided motivation through the example of vector bundles on a space, though it doesn't go that deep: http://www.cgtp.duke.edu/~drm/PCMI2001/fantechi-stacks.pdf

As for things like étale cohomology, the advice I have seen is that it is best to treat things like that as a black box (like the Lefschetz fixed point theorem and the various comparison theorems) and to learn the foundations later since otherwise one could really spend way too long on details and never get a sense of what the point is. I have certainly become a big fan of this style of learning since it can get really boring reading hundreds of pages of technical proofs.

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If you want to learn stacks, its important to read Knutson's algebraic spaces first (and later Laumon and Moret-Baily's Champs Algebriques). To keep yourself motivated, also read something more concrete like Harris and Morrison's Moduli of curves and try to translate everything into the languate of stacks (e.g. as you're learning stacks work out what happens for moduli of curves). There are a lot of cool application of algebraic spaces too, like Artin's contraction theorem or the theory of Moishezon spaces, that you can learn along the way (Knutson's book mentions a bunch of applications but doesn't pursue them, mostly sticks to EGA style theorems). Another nice thing about learning about Algebraic spaces is that it teaches you to think functorially and forces you to learn about quotients and equivalence relations (and topologies, and flatness/etaleness, etc).

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For a small sample of topics (concrete descent, group schemes, algebraic spaces and bunch of other odd ones) somewhere in between SGA and EGA (in both style and subject), I definitely found the book 'Néron Models' by Bosch, Lütkebohmert and Raynaud a nice read, with lots and lots of references too.

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 Thanks! Descent is something I've been meaning to learn about eventually and SGA looks somewhat intimidating. – Akhil Mathew Oct 20 2009 at 1:05

Underlying étale-ish things is a pretty vast generalization of Galois theory. Hendrik Lenstra has some nice notes on the Galois Theory of Schemes ( websites.math.leidenuniv.nl/algebra/GSchemes.pdf ), which is a good place to find some of this material.

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Douglas Ulmer recommends: "For an introduction to schemes from many points of view, in particular that of number theory, the best reference by far is a long typescript by Mumford and Lang which was meant to be a successor to “The Red Book” (Springer Lecture Notes 1358) but which was never finished. These notes have excellent discussions of arithmetic schemes, Galois theory of schemes, the various flavors of Frobenius, flatness, various issues of inseparability and imperfection, as well as a very down to earth introduction to coherent cohomology. Some of this material was adapted by Eisenbud and Harris, including a nice discussion of the functor of points and moduli, but there is much more in the Mumford-Lang notes." Unfortunately I saw no scan on the web.

The preliminary, highly recommended 'Red Book II' is online here.

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 Wow,Thomas-this looks terrific.I guess Lang passed away before it could be completed? Wonder what happened there. – Andrew L Jun 15 2010 at 21:47

Here is a soon-to-be-book by Behrend, Fulton, Kresch, great to learn stacks: http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1

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Is it really "Soon" though? The notes are missing a few chapters (in fact, over half the book according to the table of contents). – Charles Siegel Oct 22 2009 at 1:12
True, the project might be stalled, in that case one might take something else right from the beginning. Pity - I find the style great... – Peter Arndt Oct 24 2009 at 13:27

There are a few great pieces of exposition by Dieudonné that I really like. The first two together form an introduction to (or survey of) Grothendieck's EGA. The second is more of a historical survey of the long road leading up to the theory of schemes. I am sure all of these are available online, but maybe not so easy to find.

• Algebraic geometry ("The Maryland Lectures", in English), MR0150140
• Fondements de la géométrie algébrique moderne (in French), MR0246883
• The historical development of algebraic geometry (available here or here)
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