A learning roadmap for algebraic geometry

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)


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!


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.