Over the next few years, especially as my current students graduate, I may take on a few additional masters students, although times may come when I'll be too full (like at this current moment). 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. Because there is really no time to waste in a two-year masters program, 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 Ravi Vakil's 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 if for whatever reason you are unable to study this material in courses.) You should also be actively interested in learning about nearby subjects that interest you. Which subjects they are is up to 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. I will of course be happy to talk with you no matter whom you are working with.
My personal style as an advisor
For masters' students, I'll suggest small toy problems to think about (and these have a habit of growing into interesting serious research). You will pick one of these to work on, and I'll suggest some initial background reading to get you started on understanding the problem and its significance.
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.
I will be a demanding advisor, more demanding than most. I expect independence and that you treat your masters' studies like a job (albeit one with very strange hours depending on your natural work rhythms).
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 masters 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 current 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 our best students here at the Technion 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, and visitors. It helps to have an interesting question about their talk or one of their papers to break the ice with people you don't know personally.
On seminars:
Older graduate students will verify that there is a high correlation between those students who are doing the broadest and deepest work and those who are regularly attending seminars. Many people erroneously conclude that those who are the strongest students therefore go to seminars, while in fact the causation goes very much in the opposite direction.
Go to research seminars earlier than you think you should. Do not just go to seminars that you think are directly related to what you do (or more precisely, what you currently think you currently do). You should certainly go to every single seminar related to algebraic geometry that you can, and likely drop by other seminars occasionally too. Learning to get information out of research seminars is an acquired skill, usually acquired much later than the skill of reading mathematics. You may think it isn't helpful to go to a seminar where you understand just 5% of what the speaker says, and may want to wait until you are closer to 100%; but no one is anywhere near 100% (even the speaker!), so you should go anyway.
Try to follow the thread of the talk, and when you get thrown, try to get back on again. (This isn't always possible, and admittedly often the fault lies with the speaker.)
At the end of the talk, you should try to answer the questions: What question(s) is the speaker trying to answer? Why should we care about them? What flavor of results has the speaker proved? Do I have a small example of the phenonenon under discussion? You can even scribble down these questions at the start of the talk, and jot down answers to them during the talk.
Try to extract three words from the talk (no matter how tangentially related to the subject at hand) that you want to know the definition of. Then after the talk, ask me what they mean. (In general, feel free to touch base with me after every seminar. I might tell you something interesting related to the talk.)
See if you can get one lesson from the talk (broadly interpreted). If you manage to get one lesson from each talk you go to, you'll learn a huge amount over time, although you'll only realize this after quite a while. (If you are unable to learn even one thing about mathematics from a talk, think about what the speaker could have done differently so that you could have learned something. You can learn a lot about giving good talks by thinking about what makes bad talks bad.)
Try to ask one question at as many seminars as possible, either during the talk, or privately afterwards. The act of trying to formulating an interesting question (for you, not the speaker!) is a worthwhile exercise, and can focus the mind.
Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)
Your thesis problem may well come out of an idea you have while sitting in a seminar.
Go to colloquia fairly often, so you have a reasonable idea of what is happening in other parts of mathematics. It is amazing what can become relevant to your research. You won't believe it until it happens to you. And it won't happen to you unless you go to colloquia. Ditto for seminars in other fields.
On giving talks
There is a huge amount to say about giving talks. For now, I'll first direct you to Terry Tao, who reminds us to be considerate of our audience, and that Talks are not the same as papers.
How to give a colloquium. (First line: "Most colloquiua are bad".)
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 [2008] 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.