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!