Next summer (2020) sometime in late July or early August Wei Ho, Pieter Belmans, and I are going to have another Stacks project workshop. If you google around you’ll find the website for the previous iteration of the workshop; this one will be structured similarly. But we’ve only just started working on the organization and everything is still in flux. Feel free to email me with suggestions for things we could do, or mentors we could invite, etc. Thanks!

This semester (Spring 2019) I taught our first year, second semester algebraic geometry course for graduate students as a series of ask me anything sessions. This blog post is a report on how it worked and whether I would advise you to do the same when you teach your course.

Some background info: The enrollment was 10 students (including 4 undergraduates) but the course was attended by a few additional graduate students. Here is a link to the course webpage.

Why did I do this? First of all, I have taught algebraic geometry for graduate students many, many times and I have never taught it the same way twice. When I teach these courses, I am trying to “level up” our graduate students: I am trying to show that there is a well defined language of algebraic geometry with very sound foundations, that you can enjoy building more complicated things on top of this, that you can read about fantastic new methods/results in papers and books, and that you can do all of these things yourself actually. Maybe I can put it simply this way: I want to get them to the point where it would be fun for me (and hopefully others) to have them as graduate students. So what is more fun than a student who comes up with good questions? My second motivation was a more personal one, namely I wanted to have a bit more direct contact than usual with our first year graduate students.

How did it work? Well, I simply went to the lecture room and asked if there were any questions. After an awkward silence, usually lasting not more than one or two minutes, somebody would ask a question. Most of the time I would answer. Then another question, and so on. Often the answers were longish because of the backtracking necessary to give sufficient background in order for the answer to make sense.

Initially, the students were reading along and asked questions which allowed me to more or less develop basics of scheme theory. I tried to stimulate questions from people who preferred to think about only classical varieties, as I think this is a perfectly sound way to do algebraic geometry. Roughly in the middle of the course, in one of the lectures I limited myself to 3 minutes per answer. This lecture I think worked rather well, because I ended up getting more different questions from more different people in the room. Later I introduced another ingredient, namely, 20 questions for schemes! The students surprisingly quickly guessed I was thinking about the degree 5 Fermat surface over the field with 7 elements. I repeated this the next time, where the student, again surprisingly quickly, found that I was thinking of the spectrum of the dual numbers over the complex numbers.

An ingredient of the course which I quite enjoyed was that I was able to talk about the de Rham complex and its cohomology, both in characteristic p and in characteristic 0. Namely, the natural question arose to what extend we can find the “usual cohomology” if you are an algebraic geometer who isn’t willing to use inequalities between real numbers (such as myself). I was then also able to tie this in with coherent duality and the relationship between dualizing sheaves and sheaves of differentials.

For each of the lectures, I wrote, from memory, an account of the mathematics discussed during each meeting. You can find these on the webpage of the course linked above. As you can see, we covered an acceptable range of topics during the lectures. Also, I gave problem sets based on the topics discussed previously. (Since you do not know what the students are going to ask, you cannot give problems foreshadowing the upcoming material.) Finally, I decided to give an oral exam to the graduate students and assigned final papers to the undergraduates.

What worked well?

1. The students were amazing: they asked good questions whose answers they really desired to know.
2. You don’t need to prepare the lectures!
3. You get more buy in from your students.
4. You can cover more diverse topics.
5. Different answers can be at different levels (some of the students may already know more theory and you can occasionally discuss material which is more advanced).
6. You can react to the interests of the students in real time.
7. This structure gives you an opportunity to skip worn out paths and reorder the material drastically.
8. Students will help you prove things as you are working through them on the board.
9. I found that I was doing more examples this way. For example, we discussed the blowing up of the cone over a conic along a line through the vertex to show that blowups mayn’t be what you think they are from nice pictures in books.
10. You can skip over annoying verification without the students noticing. You may think this is a drawback, but more and more I think this is a useful and necessary evil of teaching algebraic geometry.

What didn’t work so well?

1. You don’t always remember the slickest proof of every result you talk about.
2. The order of the arguments isn’t always the right one.
3. Students may never be sure that they have completely grokked all of the material up to a certain point. The material is not presented in a linear fashion. They’ll have to do a lot of work themselves.

The last point is perhaps the most serious drawback. However, IMHO it is impossible to teach algebraic geometry and truly cover all the details needed to get a sound theory. It is necessary for those who intend to work in the field or intend to use AG in a serious manner to sit down and work/read through a good deal of the material by themselves.

Would I recommend teaching a graduate course in Algebraic Geometry this way? Actually, no. First of all, if you are teaching algebraic geometry for the first time, I recommend choosing a good text to work with and sticking fairly close to it; or choose a topic, for example linear algebraic groups and representations, choose a good book for it, and aim the development of your algebraic geometry towards the topic (maybe the book already discusses some algebraic geometry in the first chapter). On the other hand, if you have taught algebraic geometry already multiple times, then you know why it is a difficult thing to do. You probably have already found a best possible method of teaching the course for you, and there is nothing I can say to change your mind!

Also, personally, the next time I will yet again use a completely different method!

Let p be a prime number. Let k be a perfect field of characteristic p. Let U be a smooth variety over k. Choose a compactification U ⊂ X over k such that X is smooth over k and such that the divisor D = X – U is a strict normal crossings divisor D = D_1 ∪ … ∪ D_n. Then we can define the log de Rham complex Ω_X^*(log D) and try to define

H^*_{dR, log}(U) = H^*_{dR}(X, Ω_X^*(log D)

I would like to know is whether there is a published/online proof of the independence of the choice of the compactification provided one has a sufficiently strong form of resolution of singularities (RS). I did the calculation myself on a napkin (see explanation below), but it’d be great if somebody can point to a more honest writeup. Of course I searched the web for a while… also I think one of my students told me this calculation works and somebody else (maybe Illusie himself?) told me a student of Illusie worked on a better version of this a while ago? Does this ring a bell?

What do I mean by RS? Well, I mean that we can go from any compactification to any other one by a sequence of good blowing ups and good blowing downs. A good blowing up of an X as above is one which has an irreducible smooth center Z contained in the boundary (of course) such that for any I ⊂ {1, …, n} the intersection Z ∩ ⋂_{i in I} D_i is either all of Z or empty or a smooth closed subscheme of Z of codimension equal to the number of elements of I. (Aside: I think that embedded resolution of singularities will imply RS.)

Assuming RS the independence claimed above boils down to a local calculation. Think of affine r + s space A^{r + s} as the spectrum of k[x_1,…,x_r,y_1,…,y_s] with divisor D given by x_1…x_r = 0. A good blowing up looks etale locally like the blowing up of A^{r + s} in the ideal generated by x_1,…,x_{r’}, y_1, …, y_{s’} for some 1 ≤ r’ ≤ r and 0 ≤ s’ ≤ s. This blowing up is clearly equal to the blowing up of A^{r’ + s’} times the other factors. By a suitable Kunneth argument for logarithmic complexes this reduces us to the case r = r’ > 0 and s = s’. OK, so denote b : W → A^{r + s} this blowing up with exceptional divisor E isomorphic to P^{r + s – 1}. What I did was compute the cokernels of the maps

b^*Ω^i_{A^{r + 1}}(log D) → Ω^i_W(log b^{-1}D)

for all i. My napkin calculation for i = 1 showed the cokernel to be equal to Q = O_E(-1)^s. For notational convenience set S = O_E^r. Then for i = 2 my calculation gave a cokernel with a filtration having 3 graded pieces, namely

S ⊗ Q, ∧^2(Q)(1), ∧^2(Q).

For i = 3 we get graded pieces

∧^2(S) ⊗ Q, S ⊗ ∧^2(Q)(1), S ⊗ ∧^2(Q), ∧^3(Q)(2), ∧^3(Q)(1), ∧^3(Q).

And so on. If correct (caveat emptor), these cokernels have zero cohomology in all degrees. (Note that Q has rank s which is ≤ dim(E) because r > 0.) Hence the displayed arrow defines an isomorphism on cohomology and we get the desired isomorphism on logarithmic de Rham cohomology because Rb_* b^* = id on locally free coherent modules.

Looking forward to your comments!

Edit 3/16/2020. Both Ofer Gabber and Burt Totaro emailed me to say that the result is in the paper: A. Mokrane. Cohomologie cristalline des varietes ouvertes. Rev. Maghrebine Math. 2 (1993), 161-175.