This is my second report on the second semester of a yearlong algebraic geometry course for first year graduate students here at Columbia University, based on Ravi Vakil’s lecture notes. The first report is here. Please visit Ravi’s Math 216 blog and find the complete set of lecture notes here.
Besides some minor and unimportant annoyances with the text, I found teaching out of these lecture notes very pleasant. What turned out to be, for me, a key feature of his notes is Ravi’s intent to do things in the correct generality:
We will work with as much generality as we need for most readers, and no more. In particular, we try to have hypotheses that are as general as possible without making proofs harder. The right hypotheses can make a proof easier, not harder, because one can remember how they get used.
As I worked my way through the material I felt that Ravi mostly succeeded in this and it gave me the confidence to be less general! For example, I proved a bunch of results on ample and very ample invertible sheaves in the course working only with morphisms between schemes of finite type over a field. I stuck with Noetherian integral schemes whilst defining the Weil divisor class group. I talked about effective Cartier divisors, but avoided talking about Cartier divisors (a horrrible invention IMHO). Etc, etc.
Doing this allowed me to cover more ground than I usually do in an algebraic geometry course. I was able to do pushforward and pullback of divisors for finite morphisms of regular curves and prove the “n = ∑ e_i f_i” formula if you know what I mean. I was able to introduce cohomology for quasi-coherent sheaves on quasi-compact and separated schemes and actually prove some interesting theorems about it, by only doing Cech cohomology (this is probably the best time saving feature of the notes — it is one of those “why didn’t I think of that” things). Using this I was able to prove q-gr(A) = Coh(X) when X = Proj(A). A trivial consequence of these basic theorems is then the Riemann-Roch theorem in the form χ(X, L) = deg(L) + χ(X, O_X) on a projective regular curve X.
Finally, at the end I diverged from Ravi’s notes. I introduced dualizing sheaves for projective regular curves X by requiring Serre duality to hold for locally free sheaves (since there is only H^0 and H^1 you don’t need the cup product here nor Ext groups). I proved the existence of an invertible dualizing sheaf ω_X on X by first proving it for P^1 and then (using duality for a finite flat morphism) for any X by choosing a nonconstant rational functor. I defined the genus of a projective regular curve X over a field k, assuming H^0(X, O_X) = k, as g(X) = dim H^1(X, O_X). Then Serre duality gives deg(ω_X) = 2g(X) – 2. I proved that every genus 0 regular projective curve X is a conic. I proved that ω_X = Ω_X^1 if X is smooth over k (although here I had to assume something about a trace map on differentials). Finally, I explained how this leads to Riemann-Hurewitz using functoriality of differentials.
It may seem depressing to not be able to get much beyond RR and RH in a yearlong algebraic geometry course. But what really counts is for students to learn a whole language. Working through Ravi’s notes is a great way for students to do this. Thanks Ravi!