The eleventh post is the February 19 version here. (Update February 24: a revised version is now posted, see this comment.) The next post should appear on March 12. As always, I am behind on responding to emails, and to comments here. Comments from many people are incorporated (including now some from Chris Davis at UC Irvine).

For learners.

Read 15.2 and skim 15.3. Read 16.1-16.3, and look at 16.4 if you feel like it. Read 17.1-17.5. I’ve included 17.6 in case you are curious.

15.2 relates invertible sheaves to Weil divisors.
15.3 discusses effective Cartier divisors.

Chapter 16 is on quasicoherent sheaves on projective A-schemes, and graded modules. I find it surprisingly tricky, which means that it is likely harder than it needs to be, so advice is appreciated.
I intend to re-do 16.3, and give a different exposition of Serre’s theorem, and in particular introduce the notion of ampleness. I field-tested the exposition this quarter, but haven’t put it in. I hope to soon — sorry! (Update Feb. 24: now done!) I want 16.1-3 to be as readable as possible, and I’d like to leave the harder stuff that we won’t use for 16.4, which is optional. (Update Feb. 24: this is now easier in the new version, but not field-tested.)

I’ve not really edited chapter 17 since last year, so I hope it’s okay.
17.2 and 17.3 establish the basics of pushforwards and pullbacks.
17.4 is an essential application: invertible sheaves and maps to projective space.
17.5 is an essential consequence of 17.4: the “curve-to-projective extension theorem”.
17.6 (starred) discusses the Grassmannian as a moduli space. I haven’t looked at it in a while, but would be interested in hearing feedback on how accessible it is, and how to make it more accessible.

Exercises to do: there are necessarily a lot.
15.2.C, 15.2.E, 15.2.F, 15.2.H, 15.2.I, 15.2.J, 15.2.Q, 15.3.B, 16.1.A, 16.1.C, 161.E (which may be hard), 16.2.A, 16.2.B, 16.3.B, 16.3.E (Serre’s theorem — this exposition will be changed though), 17.2.A, 17.2.C, 17.3.A, 17.3.E, 17.3.G, 17.4.A, 17.4.G, 17.5.B. As always, it hurt my soul to omit some exercises from that list, so please try others that catch your eye. (In the Feb. 24 version, replace the 16.3 problems with some sampling of 16.3.C, 16.3.F, 16.3.H, 16.3.J, 16.3.K, 16.3.L, 16.4.C.)

If you are looking for exercises to give you practice with explicit examples, try 15.2.A, 15.2.D, 15.2.G, 15.2.I, 15.2.N (and lots more in 15.2), 16.2.A, 17.4.B (and lots more in 17.4), 17.4.H (fun), 17.5.A.

For experts.

15.2. The link between line bundles and Weil divisors is nonobvious and hard and takes some getting used to. I find this easy to forget.

15.3: Inflammatory comment: I’ve found very little use for Cartier divisors in this first course. Effective Cartier divisors are useful, but even they are secondary. I prefer to just stick to line bundles (invertible sheaves), and to get at them using Weil divisors. I hope I have provoked people to try to convince me otherwise.

16: I would have thought that graded modules and quasicoherent sheaves would be easy (because they are so classical), but I find them harder than the foundational formal generalities of scheme theory. Graded modules are not my friend.

16.3 will be rewritten (hopefully soon). Things like ampleness can be done in a great deal of generality, at great expense. I will do ampleness in only moderate generality, where it isn’t much work, and which suffices for all applications I’ve seen in talks. (Update Feb. 24: now done.)

Important Question: what is the “right” notion for the relative version of global generation? (“Relatively globally generated” sounds awkward.) Of basepointfreeness? (How should that last word be hyphenated?) Some will interpret this as “what is the current convention” — I’m hoping for something broader, including “this variant would be completely unconfusing, and might change the convention”.

16.4: the adjointness between functors between graded modules and quasicoherent sheaves is surprisingly annoying, and not so necessary for where we are going. Hence I’ve starred it, so people don’t think it is necessary. (My philosophy on exposition of these foundations: The straight road is not so bad, even though it passes near treacherous hills.) This has the advantage of putting the most annoying parts of 16 into an optional section, so most people can ignore it on first reading. No one liked this section when I taught from it. Interesting fact: it seems surprisingly hard for a beginner to show that every closed subscheme of a projective scheme is cut out by homogeneous equations. (Update Feb. 24: this exposition is now improved.)

17.2 and 17.3: I do pushforwards and pullbacks relying heavily on affine open sets. A lot of things are quite easy then. An interesting complication arises because pushforwards of quasicoherents are not necessarily quasicoherent, while pullbacks always are; I’ve chosen not to work in the larger category of O-modules, and to stick to affines. I think there may be errors around exercise 17.3.B, and have to dig up my (very large) to do list.

17.4: I hope I’ve foregrounded this idea enough, that in effect is the interpretation of the functor of points of projective space. Question: what is the difference between a linear system and a linear series? Because I don’t know the difference, I’ve stuck to the first. Is there any reason to consider linear systems over schemes that are not over some field? One could of course — but has anyone ever used this notion? (And if not, I’m not going to contort myself to make a definition no one uses.)

17.5 I hope the reader realizes this Curve-to-projective extension theorem is very important!

17.6 is on the representability of the Grassmannian functor. I haven’t had a chance to look at this again before posting, but feel free to comment (i.e. criticize). (Update Mar. 10 2011: I’ve taken a look, and I’m happy with it. It will be slightly tweaked in the next version, coming out around March 12.) I’d like to point out that in any introductory algebraic geometry book I know about, the Grassmannian is never defined rigorously. (Update March 8, 2011: Ulrich Gortz and Torsten Wedhorn do it properly; see my response to Torsten’s comment too.) I’d like this section to be completely rigorous, yet not too forbidding. Note that the Plucker equations are *not* used to show the Grassmannian is projective, and if you use them, life seems to be harder. (March 10, 2011: I also vaguely remember someone saying that the valuative criterion for properness is necessary to show that the Grassmannian is projective, but as you’ll see, that’s not the case.)

For everyone.

I will try to reserve “pullback” of a sheaf for pullback of a quasicoherent sheaf (f^*), and “inverse image” for f^{-1}.

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