A new version of the notes is available at the usual place (the February 21 2024 version). Everything is in potentially nearly final form for an “official version”, except for typesetting/formatting and the index. 

(a) Significant change, but not substantive:  The figures are now basically done (meaning:  redone). Because I think people should be comfortable making their own sketches in real time as they figure things out, I’ve deliberately gone all in on a hand-drawn aesthetic. This is atypical, even amateurish, for serious mathematics books. So maybe I will reconsider.

(b) About composition of projective morphisms (the old Exercise 17.3.B), and more generally 17.3:  I realize now what the complication was in the old 17.3.B.  People were stuck at many steps, but the real issue was only the last one, to get to the quasicompact target case once you already had the target line bundle. There was indeed a gap there. Given what is done in the notes (and not even by this point in the version previously posted), we can show it only in the Noetherian case.  So the new version has 17.3 seriously rearranged in a number of ways.  In particular, now the old 17.3.B is a bit later, and the argument is for when the final target is either affine or Noetherian.  (Even this requires as a black box something that will only be proved in the cohomology chapter, which is Grothendieck’s coherence theorem for projective morphisms.)    I think this is now rigorous and complete. Please let me know if there are issues.  (Some of David Speyer’s ideas also in retrospect guided me on how to improve it.)  I’ve moved all the double-starred bits to the end (and I hope the reader ignores them all).  There is a single-starred section that is where the trouble lies, and that’s going to be hard going for those readers working through it.

(c) In the chapter on the 27 lines (you know which chapter number it is), I was always unhappy about needing Castelnuovo’s criterion.  János Kollár pointed an explicit workaround that I like a lot.  (I know that, roughly, doing it in this hands-on way, is very classical; but it is hard to do it rigorously without hand-waving, and you’ll notice this hand-waving in some expositions you may have seen.)  This is now Proof 2 of Proposition 27.4.1.  His explanation to me was direct and to the point; I’ve muddied it a little to fit the narrative, so the “worsening” is due to me.  (Most mathematicians have their own particular kind of thinking, which they are best at.  Kollár has this rather amazing ability of understanding very abstract things and very concrete things, both as well as anyone else, and those two things seem to be connected directly in his head in a way that they are not for most algebraic geometers. When I was in grad school, we secretly called him “The Mighty Kollár”, and even now it doesn’t seem an inappropriate name, although I would never say it to his face.) 

General philosophical point:  I’ve noticed that there is a tension between the kinds of requests people have of the notes.     Roughly, on one hand, there is my desire to try to make it possible to cover some central core of the material in a year (for at least some people), which requires rather severe compromises.  I’m trying to make different compromises than most people have made in the past (in particular, I’d like to expect less from the reader in terms of background, but then I need to expect more from the reader in other ways).  The things I need to push back against are things like “This topic really needs to be included”, “This topic needs to be fleshed out more completely”, “This topic is not done in sufficient generality”, “This topic needs to be done more rigorously”, “The presentation of this topic is well tuned to me as a reader”, “I coudn’t solve this exercise”.  In all of these cases, the suggestions are good ones, but at some point the manuscript will sink under the weight of items loaded onto it.  Usually those making suggestions are happy to suggest what other things should be cut, but you might not be surprised to hear that these suggestions contradict each other.    I’ve tried to help by starring and double-starring some topics, but I can’t seem to stop some readers from not skipping them (and then getting boggeddown).  I’ve had a number of suggestions of things that “really are needed in such a work” that I’ve had to repeatedly decline.  Even the ones I have said yes to have let it creep up to 850 pages even after the I passed a secret “no new material” line in my mind.  (Some recent additions:  on top of the ones mentioned above:  a brief mention of projective normality; definition of Cartier divisor in generality; and more.  But even these are things I think the reader can quickly read on their own from their web having read these notes, and needn’t be here.)

One form of the compromises I’ve made is:  “If we need it, I don’t want to black-box it, and I want you to understand it, but I only need you to understand it well enough to use it and move on”, and “if we don’t need it, no matter how wonderful it is, we should just skip it, and you can learn it on your own later” (so if included, those things are starred or double-starred).  I’ve had a hard time maintaining this position consistently, and over time a lot of things have slipped by my defenses.

At this point I am still entertaining all sorts of suggestions, but am going to try to stick to things that particularly deal with mathematical errors (often leaving imperfections and imprecisions — and many of these were actually deliberate choices), or really affect the understanding of a significant portion of readers (which I have some broader sense of given comments over the years) and not just you personally.

Many of the recent comments (including some still for me to think about) are here on this website.  Some excellent ones have come to me by email, from a group of students in Poland, by way of Joachim Jelisiejew.   I want mention them here, and I also look forward to seeing what these students go on to do mathematically in a few years’ time, because they are clearly very talented.