I've been studying scheme theory from Hartshorne and Qing Liu for a few months now. (For those who are not big fans of Hartshorne, I have to note that I agree with you: I use it only for exercises.) I now have a basic understanding of concepts like separatedness and properness, and quasicoherent sheaves, although I haven't gotten to cohomology quite yet.
I do all the exercises that don't seem totally trivial (and there are indeed a few of those, in both textbooks). I do feel like this has given me a good solid understanding of the material, but I also note that it's a bit one-sided. All the exercises in both texts are pretty abstract. I do what I think makes the most of them: for each exercise, I tend to write 5-10 page solutions which carefully develop machinery to makes the problem trivial, and this has been great practice for the broad realities of research. Typically a term will come to mind that I've heard but never studied and which seems applicable, and I'll develop that concept (or what I think that concept ought to be) until it solves my problem.
At risk of repeating it too much, I'm really happy with certain aspects of what I've gotten out of this, but it's also clearly a very impoverished approach: one unfortunate result is that I rarely get to sit down with a real scheme and do some real computation and explore a real example. I'm becoming increasingly aware of the fact that if I were faced with such a situation, I wouldn't know where to begin.
So here's my question.
Hartshorne and Qing Liu have essentially the same issue. I've also used Vakil's FOAG, which is markedly better, but still a bit weak in that department. Where can I find a collection of exercises in algebraic geometry and schemes that will force me to get my hands dirty with some real schemes, and really compute something?
Best Answer
This doesn't answer your question, but: why don't you just start reading papers in algebraic geometry? You will quickly be forced to come to grips with "reality" in this way. My basic point is that, if you have read (a lot of) Hartshorne and Liu, you don't need more textbooks; you just need to start reading some research mathematics.
But regarding your actual question, you already have the answer at hand, namely: Hartshorne! Chapters IV and V are entirely about curves and surfaces, and have lots of concrete discussion of both of them. If you succeed in mastering this material, you will know a lot of concrete algebraic geometry.
A typical problem that it is hard to think about if you don't know anything is: "how do I describe a typical curve of genus $3$, or $4$"? After you read Hartshorne Chapter IV, you will know that the answers are "a smooth plane quartic", and "the intersection of a quadratic and cubic hypersurface in $\mathbb P^3$", respectively. You can't get much more concrete than that. (And there are many exercises in the spirit of such concrete questions.)
One thing is that you will need cohomology of coherent sheaves, but that it not so hard to learn, and the beautiful applications in Chapters IV and V should give ample motivation.