Math 245: Equivariant Algebraic Geometry

The goal of this class is to understand as much as possible about Riemann-Roch for Deligne-Mumford stacks, following Dan Edidin. (You can get the paper here.) This will give us an excuse to learn about equivariant techniques, Riemann-Roch, and stacks. In the first week and a half, Dan will outline the proof of Riemann-Roch for Deligne-Mumford stacks (so the punchline will come at the start), making clear what ideas need to be defined and what properties need to be proven in order to make the proof work. (I will serve as TA at this time.) In the rest of the course, we will work through these ideas.

Dan Edidin has a webpage for his notes here.

Prerequisites. You needn't --- and shouldn't --- have any background with equivariant techniques, Grothendieck-Riemann-Roch, or stacks to take this class. In fact, the goal is to learn about these ideas by seeing them in action. You will need comfort with algebraic geometry (the language of schemes), combined with a willingness to work with things you haven't fully mastered. Dan Edidin will also be around for the first two weeks, and is also happy to provide any background you might want.

Intersection theory. We will ned some comfort with the language of intersection theory for varieties/schemes. Reading Chapter 20 of my algebraic geometry notes, for example, would suffice. (We will use the construction of Fulton and MacPherson, as described in Fulton's Intersection Theory, but I can tell you what need during the course.)

Stacks. We will need the language of stacks at the start, but only enough that can be covered in a couple of hours. I gave a crash course on stacks on Tuesday January 6. Some possible references for those seeing stacks for (nearly) the first time (the first three by Edidin):