The thirteenth post is the March 31 version in the usual place. It covers up to 21.4 (although more is included). The next post should appear in three weeks, on April 23.

We have arrived in the promised land. We have now learned so much that we can reap the benefits and think cleanly about lots of things.

For learners

Read 20.4, 20.5, 20.7, the unstarred part of 20.8, and 21.1-21.4.

20.4 is on Riemann-Roch, degrees of coherent sheaves, arithmetic genus, and Serre duality. You will discover that Riemann-Roch is not hard (given what we know). We will later see that Serre duality is hard.

20.5 is on Hilbert polynomials and functions, and the notion of genus.

Skip 20.6 on Serre’s cohomological characterization of ampleness (which we mostly won’t use) unless you feel like reading it.

20.7 is on higher direct image sheaves. Like much of what we’re doing, you may find it surprisingly easy.

20.8 is on the fact that proper pushforward of coherent sheaves is coherent (in reasonable situations): Grothendieck’s coherence theorem. Read everything up until the double-starred proof of Chow’s lemma. Spectral sequences make an appearance here at the very end (of the proof of Grothendieck’s coherence theorem), and I wonder if this entire section should be starred. (At this point, I’m intending to star it in the next version.)

I hope someone tries to read the proof of Chow’s Lemma, and lets me know what can be improved! I tried to isolate the hard parts (which in some expositions can be somewhat elided, e.g. Hartshorne Exercise II.4.10).

Chapter 21 is on curves, and is a possible punchline for a year-long course. Or it can be done earlier, taking things as black boxes.

21.1 is a criterion for a morphism to be a closed immersion. It has the hardest argument (for me) in the chapter, so don’t be deterred.

21.2 collects what we know into some useful tools we will repeatedly use.

In 21.3, we apply them to curves of genus 0.

In 21.4, we apply them to hyperelliptic curves.

(If you read ahead, you’ll read about curves of genus up to 5, except for genus 1. The genus 1 — elliptic curve — case will be in the next posting, or so I hope.)

If you are interested in curves over fields that are not necessarily algebraically closed, I’d like to know if the extension to that case is readily comprehensible, or if there are steps that are harder than they should be. In particular, if you have trouble with 21.2.D, I would like to know, because the fault will be mine, not yours.

Recommended exercises: 20.4.A, 20.4.B (Riemann-Roch!), 20.4.C, 20.4.E, 20.4.F, 20.4.H, 20.5.C, 20.5.M (Bezout!), 20.7.B, 20.7.D, 20.7.E, 21.2.A, 21.2.B.

Exercises involving explicit examples:
20.5.A, 20.5.F, 20.5.S, 20.5.T, 20.5.U, 21.1.A, 21.1.B, 21.4.A.

For experts

The pushforward of coherent sheaves under proper morphisms of locally Noetherian schemes are coherent. I’m idly curious: is there a counterexample in the wacky non-Noetherian world without Noetherian hypotheses?

In Exercise 21.7.B, I want to mention the fact that there is no number N such that every genus 1 curve over a field k has a point of degree at most N over k. Is this actually known? Does anyone know a reference? On a related note, is there an easy example of a genus 1 curve with no degree 1 points (easy = they can prove it themselves)? Or failing that, is there a reasonably canonical reference? (I should presumably browse through Silverman.) [Update May 4, 2011: both are now answered, in these comments and the 14th notes comments.]

For everyone

I may have the definition of Hilbert function wrong, and at some point I will check. Currently for any coherent sheaf F, I define it as h^0(F(n)). It’s possible it should be defined only for the structure sheaf of a projective variety X, and then it shouldn’t be h^0(O_X(n)), but the dimension in the image of H^0(O_X(n)) of H^0(O_{P^N}(n)). I will check. (Or if someone can immediately point me to some reference, that would be great…)

I have some notes on defining the cup product in Cech cohomology. This is a bilinear map H^i(F) x H^j(G) –> H^{i+j}(F \otimes G) that is associative, and “symmetric” (up to a sign). This is something that I’d not otherwise learned. Does anyone feel this is worth including? Is the cup product in cohomology something that people use often enough to make this worthwhile? The fact that I haven’t included it in this draft shows that my default answer is “no”: I’m trying to find excuses not to include things.