# 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.

• The course should start from the beginning, and require little background. Traditionally students are thought to need a lot of background before they could even start learning. I don't think that's necessary; I don't think there is time for that (unfortunately) in a graduate program; and I think requiring too many prerequisites deters people in nearby fields --- they don't have the time, and they also think scheme theory is harder or more technical than it is. In particular, the course requires little commutative algebra in advance. (Indeed, it is very helpful to learn your commutative algebra at the same time as you learn your algebraic geometry, just as it is very helpful to learn your algebraic geometry at the same time as your commtuative algebra.)
• The course should not be excessively "formal", and instead should have as much geometry and as many examples as possible. Students should have time to develop geometric intuition for the ideas they learn.
• The course should cover as much as possible of the canon (ideally all).
• The course should be complete. There is a well-intentioned tendency in algebraic geometry classes to leave out the key proofs (including facts from commutative algebra or homological algebra), by referring to other texts. Very few people follow up and read those other references. Central insights get lost. So I've tried to omit as little as possible, and to have anything omitted included (as "starred" sections) in the notes. That way students can see that things I omit take only half a page to prove, and that they shouldn't be scared.
• The course has lots and lots of exercises. Many of them would be considered "trivial" by experts. I don't see this as a bad thing.
• The course should be accompanied by readable notes that can be reviewed by students afterwards. Otherwise, it is much harder to learn. Readability should be measured by student response: can they do the exercises? Do they ask specific questions?
To try to manage this, brutal decisions must be made:
• No further goals are allowed.
• There is no room in a very full course to add more material without either cutting other material, or rushing and making the material incomprehensible. In particular, the course is purely algebraic, and ignores analytic aspects. (We should have complex analytic courses every other year.)
• On a related note, there is no time to start with varieties. To compensate, varieties are the central example throughout, and we have to work hard to keep things tied down to earth. But I think that more can be done by having a continuous year-long course rather than having half a year on varieties, and half a year on schemes. And this can be done without any loss in student numbers. I realize this is likely specific to Stanford.
• Students must do homework. A lot of it. But the upside is that I've been really really impressed by what they do --- I can't emphasize this enough. (And I also can't emphasize enough that the people doing hard homework aren't just the future algebraic geometers. I have been very very impressed by people in other fields, and even non-neighboring fields.)
• More generally, students have to work hard.
• The course has to move very fast. But it can't move too fast or else it will be useless. This can be empirically tested: if students can't do the exercises (after lots of thought and work), then the course is going too fast.
• At the start, some time has to be spent on thinking categorically, and then understanding sheaves. Students do not have facility with this coming in (as a strong rule). It can't be over-rushed. One can't just state the theorems and proofs; it is essential that the students develop insights. But this is a good thing; if they get nothing else out of the class, they will have developed ways of thinking that are central to much of mathematics, and the benefits can last a lifetime.
• The cost that hurts me the most is that it takes so long to get to fun "punchlines", answering interesting questions. Once the material is fully assembled, I can likely ameliorate this a bit. But I see no real way around it.
There are some decisions that may be idiosyncratic. For example:
• I don't restrict to things finite type over an algebraically closed field. If you are learning schemes, even if you are interested in working in "geometry" (finite type things over an algebraically closed field), I think that restricting needlessly to this setting can make things harder for a number of reasons.
• In learning new concepts, some people prefer to hear something in as general a form as possible, and then specialize. I prefer to teach in the other direction: to understand something concrete, and then to abstract from it.
• Some specific pedagogical choices: commutative algebra is developed simultaneously with the geometry. Varieties are a constant central example. We think categorically throughout. Valuative criteria are mentioned, but the hard direction is never used (in the class), so it isn't proved. Derived functors are used to prove almost nothing about cohomology that we actually use (in the class), so they are done later, because students find it harder. I can say more about the reasoning behind these choices.

(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:
• "I found these errors: ..."
• "I found these typos: ..."
• "Topic X should certainly be learned in a first year of algebraic geometry. Why didn't you include it?"
• "Here is a great explanation of this theorem."
• "Your explanation of this idea was confusing than it needed to be." (And possibly: "Here is a much better [or different] way of explaning that.")
• "Here is a cool example I wish someone had told me when I was younger."
• "My students had a hard time with this notion."
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.