Over the next few years, I may take on a few additional Ph.D. students, although times may come when I'll be too full (e.g. a time that ended recently). This page is intended for those considering working with me, although it also contains some tips for graduate students in general, as well as an idea of what I expect.
Algebraic geometry (or at least my take on it) is a technical subject that also requires a good deal of background in other subjects, as well as geometric intuition. So before I take you on as a new student, you should be comfortable with the foundations of the subject, which means having done the majority of the exercises in Hartshorne or my course notes, and being able to explain them on demand. (You shouldn't do this on your own; I'm happy talking with you through this process.) You should also be actively interested in learning about nearby subjects that interest you. Which subjects they are is up to you. If you're not interested in regularly attending talks, and being broadly interested in mathematics outside of your thesis topic, or if you don't feel like getting technical in a rather serious way, I'm probably not a good fit for you.
If you are interested in some of the ideas of algebraic geometry, you should also consider a number of other advisors. In this department there are a good number of people interested either directly or indirectly in algebro-geometric ideas. You can read about them here. I will of course be happy to talk with you no matter whom you are working with.
My personal style as an advisor
I'll suggest problems to think about, starting from small toy problems (which have a habit of growing into interesting serious research). You'll have to pick what to work on, and find your own thesis problem. Mathematics isn't just about answering questions; even more so, it is about asking the right questions, and that skill is a difficult one to master.
I like to meet my students every week (except for exceptional weeks, of which there are many). You may prefer not to meet in a given week if you have nothing much to report, but those weeks are particularly important to meet.
The disadvantage of being a student of a young parent is that you'll have to be prepared to be more independent.
I will be a demanding advisor, more demanding than most.
I have pretty broad interests in and near algebraic geometry. To get an idea of the things I think about, see some of the things I've written. However, some of those subjects may not be ideal for a Ph.D. student for a number of reasons. I'm interested in lots of things. I may however not be the ideal person to supervise lots of things. For example, I will not supervise a thesis in a nearby field. But I definitely do not require that you work on problems directly related to my own research.
General advice (which would apply particularly to my own students)
Think actively about the creative process. A subtle leap is required from undergraduate thinking to active research (even if you have done undergraduate research). Think explicitly about the process, and talk about it (with me, and with others). For example, in an undergraduate class any Ph.D. student at Stanford will have tried to learn absolutely all the material flawlessly. But in order to know everything needed to tackle an important problem on the frontier of human knowledge, one would have to spend years reading many books and articles. So you'll have to learn differently. But how?
Don't be narrow and concentrate only on your particular problem. Learn things from all over the field, and beyond. The facts, methods, and insights from elsewhere will be much more useful than you might realize, possibly in your thesis, and most definitely afterwards. Being broad is a good way of learning to develop interesting questions.
When you learn the theory, you should try to calculate some toy cases, and think of some explicit basic examples.
Talk to other graduate students. A lot. Organize reading groups. Also talk to post-docs, faculty, visitors, and people you run into on the street. I learn the most from talking with other people. Maybe that's true for you too.
Here's a great story from Mark Meckes that simultaneously illustrates a number of points. By chance, I recently saw a PhD thesis whose acknowledgements ended with the sentence "Finally, I would like to thank Dr. Mark Meckes, whose talk in Marseille in May of this year  provided the final insight I needed to completely answer Kuperberg's Conjecture." What is interesting about this is that not only had I never heard of Kuperberg's Conjecture, but my talk was completely unrelated to the subject of the thesis, and even after reading the relevant section of the thesis I still couldn't see the connection. So one truly never knows where useful insights will come from. One of the many things I love about this story is that I don't find it at all surprising! So go to talks --- and give talks --- and talk to people!
When thinking about advisors, talk to past and current graduate students. (My former and current students: Eric Katz 2004, Rob Easton 2007, Andy Schultz 2007, Jarod Alper 2008, Joe Rabinoff 2009, Nikola Penev 2009, Jack Hall 2010, Dung Nguyen 2010, Atoshi Chowdhury, Yuncheng Lin, Daniel Litt. I also collaborated with Kirsten Wickelgren 2009, who worked with Gunnar Carlsson.)
Advice from others:
Specific advice about algebraic geometry at Stanford
Sign up for the algebraic geometry mailing list.
Go to the Western Algebraic Geometry Seminar, a twice-yearly conference.
Occasionally go to Berkeley when you hear about something particularly interesting.
When you are up to it, subscribe to the daily mailing of abstracts of algebraic geometry papers posted to the arXiv. Then most days, just delete them, but when you have some time, browse through them, and read the abstracts that catch your eye. You'll gradually get a sense of what is going on in the field. Caution: this can be psychologically damaging, as you'll feel "here I am stuck on this simple problem, and thousands of papers are coming out...". So only do this if and when you're ready. I might delete this paragraph at some point if I realize it is counterproductive.