A revised version is now posted at the usual place. The first quarter of our academic year is now over (meaning I’ve completed the first third of the course), so now is a good time to report how things went.

First, I’ll briefly explain changes since the last version (which was posted without much fanfare late in the quarter). (i) The long-promised starred section on geometrically connected (and irreducible and reduced and integral) is now added (10.5); if anyone tries any of it, please let me know how it goes. (ii) The section on (very) ample line bundles was pulled out of 16.3 (on globally generated and base-point-free line bundles) into a new section (17.6) because Giuliano Gagliardi pointed out that it used pullbacks Important Theorem 17.4.1 (describing maps to projective schemes in terms of line bundles — also known as the functorial description of projective space). Odds are unfortunately high that I’ve screwed up some dependencies, so if you find yourself in that part of the notes, please keep an eye out for unintended consequences. (iii) I’ve finally changed (open, closed, locally closed) “immersion” to (open, closed, locally closed) “embedding” throughout (the horror!).

Now back to a report on the class. I was pleased with how it went — we covered more than I’d hoped, reaching 12.2. This is far more than a third of the notes, even taking into account the chapters and sections that are not yet fully written. I should admit that the class was unusually strong this year, and thus not representative, but my hypothesis that one can cover a huge amount of the foundations of the subject in a single-year course looks like it might be vindicated. There were fewer comments in class than usual (has the exposition solidified too much?), but the problem set solutions were very strong. Partially by design, we ended with a “punchline” (a pleasant concluding topic that I think courses should ideally have — even if it is often not possible): I spent much of the final lecture discussing lines on surfaces in 3-space. Given what we know, we were able to prove interesting things, and also get a glimpse of geometrically interesting ideas in the future. I hope this helped give some relief from the intense barrage of formalism leading up to it.

I have two possible punchlines in mind for the second (winter) quarter: the theory of curves, or intersection theory. And while it is dangerous to predict two quarters in advance, one possible punchline for the final (spring) quarter is the topic of the 27 lines on “the” cubic surface.

Comments, suggestions, and corrections many of you have sent in (by commenting here, or by emailing me) have been very helpful. I am behind on responding to them, but I’m notably less behind than I was at the start of the quarter, and I expect this to continue. So please keep sending in comments (even highly opinionated ones)!

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