Math 216 (Foundations Of Algebraic Geometry) Course Notes

By the end of the 2009-10 academic year, I hope to have a reasonable version of my course notes for Math 216, a class I've taught a few times (3 times officially). As of the end of March 2010, over 2/3 is in good shape (as the course is 2/3 done). The notes distill things people have told me through the years, of slick explanations of the basic things one needs to know to work in algebraic geometry. The parts written are suitable for private distribution, and I'd now like to refine them further, by digesting into it more collective wisdom. (This webpage was written while I was bored on a long flight, without much forethought, so it is rough around the edges.)

Before I start, I should say up front that there are a number of possible good "first algebraic geometry" classes, e.g. complex algebraic geometry, or varieties, or an examples-based course, or Riemann surfaces, or arithmetic geometry, etc. This is only one approach, and we need them all.

The point of the class

There is a certain body of material one should know in order to work in algebraic geometry. On the algebraic side, this includes the basic theory of schemes. I have found the consistency of what people think new researchers need to know (to follow developments in the algebraic side of the field) to be surprising. People working in all parts of the subject (including arithmetic, commutative algebraic, etc.) seem to agree to a remarkable extent. So there remains a canon in the subject, and we have not splintered into many disconnected subfields.

This material takes a full year (more than an academic year) of sustained hard work to learn.

But most people seem to have found this material more difficult than it needs to be, and I think there are a number of reasons for this. In any given course, there is a temptation to leave things out, and refer people to other sources. (Few people follow up and read these sources; they are too busy keeping up with the class.) There is a temptation to give exercises which are hard enough to deter all but those few who are simultaneously particularly smart, prepared, and persevering.

Furthermore, I'd like to think that algebraic geometry is a central subject in pure mathematics, and that its insights should be useful to those working in other subjects. I'd like to think that many of the best people in nearby fields (number theory, topology, geometry, commutative algebra, ...) would want to have a good understanding of these insights. A good algebraic geometry class should welcome those students, and not be aimed just at future algebraic geometers, without the material being excessively watered down.

So my goals in this class were the following.

To try to manage this, brutal decisions must be made: There are some decisions that may be idiosyncratic. For example:

(You can read more about the point of the notes in their introduction, available at the top of the course webpage.)

What I want to do now (say, in the 2010-11 academic year)?

How I think about these foundational questions comes from explanations people have told me. There are often slick explanations that are passed around by word of mouth rather than in writing. So in the next year or so, I'd like to run this by a broader audience, and get feedback, along the lines of: So it would be cool if someone were willing to use part of these notes as a text or supplementary text for their class; or tried teaching from it; or supervised a reading course out of part of them; or had some students try to learn from it on their own. Or even just browsed through them and sent in ideas. I would really appreciate it.

I fully appreciate that Stanford students are not typical, and that these notes are not suitable for most people.

I'd be happy to send paper copies (for you and your students).

Some questions people have asked

What is this? Are you writing a book?

I'm not sure what I'm doing, beyond trying to aim a course at a well-defined group of actual students. The notes might naturally make a book, but very possibly not, because there are disadvantages in setting things down in stone. Thinking about it as a potential book can lead me to wrong decisions; I want to please my students, not some other constituency. (Also: do we really need yet another algebraic geometry book?) It might be posted on a webpage in pdf format. It might circulate privately as photocopied notes. But I'm not sure. I've been chewing over this course for quite a few years, and have found it helpful to think no more than a year ahead.

Why don't you post a pdf?

There are many reasons, and here are a few. (i) For the same reason, I don't give out drafts of my papers until they are suitable for public distribution: it is very much a work in progress, and if and when an electronic copy circulates, I want it to be one I am happy with. (ii) I don't feel guilty about this, because I've made available essentially the same exposition on webpages for earlier versions of the class, e.g. here. But David Eisenbud told me that posting a pdf is not high risk, so at some point in some way I will do it.