Math 216

Please take a look at Ravi’s blog about his Math 216 graduate course at Stanford university. Students and others have been chiming in leading to a total of 265 comments in 6 months. A group of people (mainly graduate students?) are working through the material as it gets updated on Ravi’s blog, and these people provide most of the which are helpful and constructive comments on the blog. Moreover, even though Ravi is not actually teaching his course this year, the blog gives one a sense of activity much like for a real course. Of course, since I am teaching my algebraic geometry course this year based on Ravi’s lecture notes, I may be more inclined to say so than others.

You can download the latest version of Ravi’s notes here. Let me give you a bit of my own preliminary impression of these notes; you can read Ravi’s philosophy behind them on his blog and in the introduction to the notes.

As everybody who has taught an algebraic geometry course knows it is virtually impossible to feel satisfied with the end result. In my experience it actually works well when younger people teach it because they have a fresh take on it, want to get to some particular material that is important to them and they are less likely to get stuck in details. I personally never teach algebraic geometry the same way twice, and I usually end up covering a fair amount of material despite feeling like I did not at the end of it.

One of the pleasing aspects of teaching the material out of Ravi’s notes is that I do not have to organize the material as much as I usually do. Mostly I am happy with the order in which things get done, although I moved the material on quasi-coherent O_X modules and on morphisms of schemes earlier in my lectures. Also, in hindsight, I should probably have skipped chapter 2 (category theory) and jumped straight to the chapter on sheaves. A key feature of Ravi’s notes is that more than 75% of the proofs of lemmas, propositions, and theorems are left as exercises. As lecture notes often Ravi explains why things are true, with lots of examples, rather than providing a formal proof. Results from previous exercises are used throughout the text, not always with explicit references (especially in the exercises themselves of course). When lecturing it sometimes made me wonder to what extend I’ve really built up the theory from scratch (which is the stated goal of the course). Of course here you can rely on outside references and ask that students read those, ask that the students do lots of exercise, and so on. One of the standing assumptions underlying the setup is that students will work hard on their own to understand the material. Moreover, I think no matter how you teach algebraic geometry you cannot build it up completely from scratch in your lectures, i.e., the students are always going to have to do a lot themselves, and maybe by building it into the course material they are more likely to do it?

Is it a good idea to have many different algebraic geometry texts? Tentatively, I would say more is better. I have personally found Ravi’s notes useful in the following way: if you can find what you’re looking for in Ravi’s notes (e.g. by googling) then you’ll quickly find pointers unencumbered by details or generalities.

Overall I am very happy with my course and the notes so far. One of my questions is how much commutative algebra I will cover teaching the course in this way (traditionally at Columbia we teach a first semester of commutative algebra and then a second semester on schemes — in one semester focused entirely on commutative algebra you can cover quite a bit). I’ll report on this in another post about Ravi’s notes at the end of the next semester, so stay tuned.

More projects

Here is a list of projects that make sense as parts of the stacks project. (For a list of algebra projects, see this post.) This list is a bit random, and I will edit it every now and then to add more items. Hopefully I’ll be able to take some off the list every now and then also. If you are interested in helping out with any of these, then it may be a good idea to email me so we can coordinate. It is not necessary that the first draft be complete, just having some kind of text with a few definitions, some lemmas, etc is already a good thing to have. Moreover, we can have several chapters about the same topic, of different levels of generality (the reason this works well is that we can use references to the same foundational material in both, so the amount of duplicated material can be limited).

  • If X is a separated scheme of finite type over a field k and dim(X) ≤ 1 then X has an ample invertible sheaf, i.e., X is quasi-projective over k.
  • If f : X —> S is a proper morphism of finite presentation all of whose fibres have dimension ≤ 1, then etale locally on S the morphism f is quasi-projective. This also works for morphisms of algebraic spaces.
  • Local duality; see also the corresponding algebra project.
  • Cheap relative duality for projective morphisms. Start with P^n over a (Noetherian) ring and deduce as much as possible from that.
  • More on divisors and invertible sheaves, Picard groups, etc.
  • Serre duality on projective varieties.
  • Classification of curves.
  • Quot and Hilbert schemes.
  • Linear algebraic groups.
  • Geometric invariant theory. I think that a rearrangement of the material in the first few chapters of Mumford’s book might be helpful. In particular some of the material is very general, but other parts do not work in the same generality. Note that we already have the start of a chapter discussing the myriad possible notions of a quotient, see groupoids-quotients.pdf.
  • Resolution of two dimensional schemes.
  • Semi-stable reduction theorem for curves. (Is there any way to do this without using resolution of singularities of two dimensional schemes or geometric invariant theory?)
  • Abstract deformation theory a la Schlessinger (but maybe with a bit of groupoids thrown in).
  • Deformation theory applied to specific cases: zero-dimensional schemes, singularities, curves, abelian varieties, polarized projective varieties, coherent sheaves on schemes, objects in the derived category, etc.
  • Brauer groups of schemes.
  • The stack of curves and pointed curves, including Kontsevich moduli stacks in positive characteristic are algebraic stacks.
  • The stack of polarized projective varieties is an algebraic stack.
  • The moduli stack of polarized abelian schemes is an algebraic stack.
  • The stacks of polarized K3 surfaces.
  • Alterations and smoothness (as an application of moduli stacks of curves above).
  • Add more here as needed.


Just a short update. The semester is in full swing here at Columbia University and there are a lot of things to do (including writing letters of recommendation), so I have had less time to work on the stacks project. I hope/expect to get back to it soon.

Currently, I am still working through the details of the paper by Raynaud and Gruson. I found a (repairable) error in the proof of the main geometric result (existence of devissage; last sentence of the proof of Proposition 1.2.3). It is a small error, but it really is an error and you have to slightly change the set-up in order to fix it. Of course I may be wrong, but I do not think so (for those of you who are taking a look at the paper: try to imagine what it would mean to replace the sentence mentioned above by a fully written out argument, checking all the details). In addition to this, I’m having trouble finding simplifications for almost any of the arguments, as each of the later results in the paper uses the earlier results, in other words, I haven’t been able to split off some parts as independent from the rest.

I am going to finish writing it all up, as soon as I have more time. But for the moment this experience is teaching me a lesson. Namely, I started working through the details of Raynaud-Gruson as I wanted to have a very general result on flattening stratifications. I was eager to do this, as I wanted to discuss Hilbert schemes/spaces/stacks in the “correct” generality. And this in turn I wanted to do because I want to explain the proof of Artin’s result that a stack X in groupoids over (Sch) whose diagonal is representable by algebraic spaces such that there exists a surjective, flat, finitely presented morphism U —> X where U is a scheme is an algebraic stack. Looking back what I should have done is write a chapter on Hilbert schemes/spaces parameterizing finite closed sub schemes/spaces/stacks (maybe even restricting the discussion to the representable separated case). This is much easier, is quite interesting in its own right, and is sufficient for the application in the proof of Artin’s theorem.

On the upside, I have learned a lot more about flatness in the effort to get this material written out fully!