Math 216 (Foundations Of Algebraic Geometry)
Course Notes
By the end of the 200910 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 examplesbased 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
wellintentioned 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
yearlong 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
nonneighboring 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 overrushed. 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 201011 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
welldefined 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.