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A little bit vague, but I hope useful for the entire community.

I am, by training, an analytic number theorist. I have managed to learn some algebraic geometry, by reading parts of Silverman's Arithmetic of Elliptic Curves, the beginning of Ravi Vakil's notes, and a very little bit of Hartshorne (and doing a least a few of the exercises).

It has become apparent to me that it would be very useful to learn more -- the subject often comes up in reading, seminars, and discussions with colleagues, and besides I find what little I know to be fascinating.

Unfortunately I don't have the time to, say, read Hartshorne (or EGA) and do all the exercises. That said, how would MO readers recommend that I go from knowing a little bit of algebraic geometry to a modestly bigger bit of algebraic geometry?

My goals, roughly speaking, are:

  • To have a broader base in the subject, and understand the basic definitions and examples better. (Naturally I am especially interested in the side of the subject that relates to arithmetic, and less so in the complex geometric side.)
  • To be better prepared to read books related to some of my research interests and those of my collaborators (Borel's Linear Algebraic Groups, Mumford, Fogarty, and Kirwan's Geometric Invariant Theory, etc.)
  • To better know where to look if I run across an algebro-geometric argument in the literature.
  • To enjoy myself, especially since I hope to persuade colleagues and grad students to join me.

There seem to be various books which are well-suited to this, e.g., Mumford's Red Book, Eisenbud and Harris's Geometry of Schemes, Harris's Algebraic Geometry: A First Course, and many others. Vakil's notes are also excellent, but perhaps for someone with ambitions of learning the subject more thoroughly. For the most part, I'm not very familiar with the virtues and drawbacks of these books.

Can MO readers recommend a roadmap for learning at least a little bit more about this subject, even as I am obliged to keep most of my attention elsewhere?

Thank you very much!

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    $\begingroup$ Qing Liu "Algebraic Geometry and Arithmetic Curves" perhaps. I am not well placed to answer but since you did not metion it and it seems a very natural choice I thought I mention it. It has an arithmetic slant is rather recent and I heard it said (technically I read it, here on MO of course ;)) that some things that are excercises in Hartshorne are proved there as results. So, it could be the easier read. $\endgroup$
    – user9072
    Commented Dec 9, 2012 at 17:48
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    $\begingroup$ I don't think algebraic geometry comes in «modest amounts»... :-) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Dec 9, 2012 at 17:54
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    $\begingroup$ Although for some arithmetic applications, you would need schemes, learning about (quasiprojective) varieties doesn't take a huge effort and is a good place to start. The first part of Mumford's Redbook does this, and also Shafarevich's Basic Algebraic Geometry. I think also that Milne has some notes on website which might be worthing looking at. $\endgroup$
    – Donu Arapura
    Commented Dec 9, 2012 at 18:25
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    $\begingroup$ This is only a narrow slice of algebraic geometry, but I recently had to learn some etale cohomology (in order to count connected components of varieties, and thence to count points on such varieties over finite fields, which is sort of an analytic number theoretic-ish thing to do), Milne's notes and Szameuly's book were very helpful (with the latter being particularly good at connecting things to both Galois theory and the topology of manifolds). And SGA wasn't actually that scary either, once one took the plunge... $\endgroup$
    – Terry Tao
    Commented Dec 9, 2012 at 18:37
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    $\begingroup$ I second the recommendation of Mumford's Red Book. It's very well written, and it focuses more on etale morphisms and less on coherent cohomology (the latter of which is not as relevant to you). For schemes, I also recommend short sets of notes on schemes, such as Szamuely's renyi.hu/~szamuely/galschemes.pdf, which will give you the basics of schemes. $\endgroup$
    – David Corwin
    Commented Dec 9, 2012 at 19:13

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I'm not even a number theorist, but when I was in grad school I took a one-semester course from Hartshorne, using his book. I felt that it carried a powerful message, more abstract and more general than what I needed for any purpose, but still really interesting. We only made it to the beginning of the third chapter, but that was enough. In fact, without entirely realizing it, I was taking a lot of hard commutative algebra results on faith in the homework, but somehow it was okay.

So I would ask whether Hartshorne is exactly what you want; it exactly explains what you say is missing from your experience. You don't have to and shouldn't read it page by page, confirming every assertion. You can learn from this book using the spiral method. I do not know "Geometry of Schemes", although a book with that title and those authors sounds good too. What I can say is that Hartshorne is especially enthusiastic about Grothendieck's perspective on algebraic geometry. Again, even though I had no career reason to care, I ended up wanting to prove things Grothendieck style: using schemes, with the same argument in characteristic 0 and positive characteristic, with proper schemes as a replacement for projective varieties, etc.

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    $\begingroup$ It would be interesting to know how many people got past Hartshorne's third chapter. I'd guess the proportion to the number of people that studied the book is quite small! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Dec 9, 2012 at 18:19
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    $\begingroup$ Chapters 2 and 3 are the "best" parts of the book. They are very abstract, some would say excessively so, and in that way not really a complete view of the field of algebraic geometry. The other chapters have a different style, and I guess serve to bring you back to earth. Anyway, it was a unique experience and I enjoyed it. $\endgroup$
    – Greg Kuperberg
    Commented Dec 9, 2012 at 18:33
  • $\begingroup$ +1. Chapter II and III are the heart on the book, and really excellent. I think it is by far the best choice for what you ask. As I understand it, you need to feel comfortable when people tells you about schemes aver a basis, proper or smooth or étale morphisms, coherent sheaves, sheaf cohomology, etc. Hartshorne will provide you all, wiht of course some efforts of your part but I don't think this difficult theory can be learnt effortlessly. The one thing that is missing in Hartshorne and that you are likely to meet is the functor point of view in AG. For that you can read the intro of EGA I. $\endgroup$
    – Joël
    Commented Dec 10, 2012 at 15:18
  • $\begingroup$ Note that the "functor point of view" is also fairly well explained in Mumford's Red Book. (And the same explanation, word-for-word, appears also in Lecture 3 of Mumford's Lectures on Curves on an Algebraic Surface.) $\endgroup$
    – Charles Staats
    Commented Dec 11, 2012 at 0:46

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