I should say up front that there are a number of possible good “first algebraic geometry” classes:
- varieties
- Riemann surfaces
- a non-rigorous examples-based course
- exposure to big ideas without complete proofs
- complex algebraic geometry (with analytic methods)
- algebraic geometry for number theorists (perhaps focusing on elliptic curves)
- and more.
We need them all. But we can’t have them all in one class. There is no royal road to algebraic geometry. This is intended to be only one way in.
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 algebraic geometry has 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 are the following.
- The notes 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 notes should cover as much as possible of the canon (ideally all).
- The notes 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. It is also common to use phrases such as “with more work, one can show that…”, or to leave serious theorems to hard exercises that most readers won’t be able to do. Central insights get lost. So I’ve tried to omit as little as possible, and to have anything omitted included (as “starred” or “double-starred” sections) in the notes. That way readers can see that things I omit take only half a page to prove, and that they shouldn’t be scared (without necessarily reading the proofs).
- The notes have lots and lots of exercises. Many of them would be considered “trivial” by experts. I don’t see this as a bad thing.
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 notes are purely algebraic, and ignore analytic aspects (except in passing remarks).
- 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. (This may be specific to Stanford.)
- Readers must do homework. A lot of it. But the upside is that I’ve been really really impressed by what people have done in the original classes — 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, readers have to work hard.
- The notes have to move very fast. But they can’t move too fast or else they will be useless. This can be empirically tested: if readers can’t do the exercises (after lots of thought and work), then the notes are 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/functorially throughout, but try to explain why this is the right thing to do. Valuative criteria are mentioned, but the hard direction is never used in a logically necessary way. Cech cohomology can be used to prove almost everything about quasicoherent cohomology that we actually use (in the notes), so derived functors are done later, because students find it harder. I can say more about the reasoning behind these choices.
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