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!
Best Answer
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.