This is for students who are thinking about doing a Ph.D. with me. 

I’m working on algebraic geometry, which is generally considered to be a subject requiring substantial preparation. What I usually suggested a student to start is by reading a book on algebraic curve and one on commutative algebra (e.g. Atiyah-Macdonald). After you get a taste of the flavor, I expect you to finish Hartshorn’s book by yourself (that includes solving most of the problems there). This should be done if one would want to work on any branch of algebraic geometry.

To know what I’m working on, which I usually call higher dimensional geometry, you can take a look at the papers I wrote. I have a broad interests on this topic. If this is what you like, then the book by Kollár-Mori is a must. I also highly recommend Kollár’s other books and Lazarsfeld’s Positivity in Algebraic Geometry I & II. Of course you don’t have to read all of them before you start your own research.

I will usually point out to you a direction that I consider interesting and potentially workable, and I expect the best students to discover their own problems on based on the materials  they read. Occasionally I would suggest a baby question, which itself could serves as a weak thesis, or even turn into a more interesting one. But I do believe ambitious students should really learn how to pick up their own problems. Indeed, finding yours own problems is one of the most important talents of a researcher. I also believe being independent will benefit you tremendously in the long run.

I have just started to be an advisor not long ago, which means I’m still shaping my strategy of advising students.  

I also suggest you to look at Ravi Vakil’s page on this matter.