A new version of the notes is available here. (It, and all older versions, are available at the usual place.) I don’t have much to report, but I just wanted to post the current version as it continues to converge. Please do continue sending in comments, particularly in the next month. There is a good chance that fairly soon a “snapshot” will be taken for an official publishable version (although that doesn’t mean that it will then be frozen).
In some more detail: I’ve now implemented the vast majority of the ideas from suggestions people have given, although there are some still to go. I’m redoing the figures in a consistent style, and about 2/3 of them are redone so far. The index will get some attention later, but the raw material is there, and I welcome suggestions and corrections. The only things I’m not interested in are latex issues (margins, etc.; but typos and errors in spacing are fair game). I am now ready for comments on figures, although there are still 1/3 of them that I haven’t re-done.
I’ll very be busy with other duties for the next four years, so I will be able to spend much less time and attention on this. I suspect it will be enough of a break that I won’t be able to return afterwards with everything as fully in my head as it is now, so this will likely be my last chance for me to really make this as good as it can be before letting it go off into the world on its own. This really feels like nearing the endgame.
So thank you all for accompanying me on this stage of what has been a most interesting and rewarding journey for me.
Math 216: Foundations of Algebraic Geometry 
February 6, 2024 at 1:17 pm
Hi Dr. Vakil,
I hope you’re well. Just a typo, I think:
In the displayed equation in Exercise 28.3.B.(b), on p. 766: should the $\frac{\partial^2 f}{\partial x \partial y}$ on the left side of the inequality be squared? (Isn’t this supposed to be the condition that the Hessian is nonvanishing?)
All the best, Ben
Ben Blum-Smith, PhD (he/him)
Postdoctoral Fellow
Department of Applied Mathematics and Statistics
Johns Hopkins University
February 6, 2024 at 4:50 pm
Good catch, thanks! Now fixed.
February 6, 2024 at 5:56 pm
I’ll take the opportunity to advertise: I’m currently teaching out of your book, and I’m blogging about it at https://sites.lsa.umich.edu/math632/ . I had thought that the blogposts would be more directly about the book, but most of them so far are about issues which have come up in class.
You might be interested in the most recent post: https://sites.lsa.umich.edu/math632/2024/02/06/the-confusing-issue-about-maps-to-projective-space-over-a-base/ , where I explain how our class was very confused about the relative version of Theorem 15.2.2, but I eventually conclude that your statement was exactly right.
February 6, 2024 at 6:31 pm
Thanks! It is interesting to watch that. Also in my (now very small) folder of things to still reply to is an email from you about this very course!
February 8, 2024 at 8:02 am
A note on the index: You currently have an entry: “principal divisor, see principal Weil divisor; principal effective Cartier divisor”.
But “principal Weil divisor” says “see Weil divisor,
principal” and there is no entry for “principal effective Cartier divisor”, only for “effective Cartier divisors, principal”. Perhaps you should simplify these references to point to the final destination?
Also, there is an index entry for “effective Carier divisors”; the middle word is missing a “t”.
February 9, 2024 at 11:16 am
All fixed! (The index is the last major thing which still needs a lot of work of course. But I’m happy to move it forward. For some reason the page numbers seem sometimes not right; perhaps I need to repeatedly latex and index even more than I do.)
February 8, 2024 at 10:20 pm
Just noticed a very minor thing that’s been there for a while: p. 110 last line before 3.2.10 – “Spec of ℤ” should probably just be “Spec ℤ.
February 9, 2024 at 10:59 am
That was actually intentional: I’m just saying in english (somewhat informally) Spec of the ring “Z localized at …”.
February 9, 2024 at 12:55 pm
Oh, right, sorry about that.
February 10, 2024 at 7:09 am
no need to be sorry!
February 9, 2024 at 10:51 am
Hi, professor Vakil! Thank you very much again for the great book! Here are some more issues I’ve come up with after the previous batch, along with things I think probably not fixed in the previous batch:
(Previously posted)18.8.3: The usage of m_p is inconsistent with the definition in the index, 2.1.1, where m_p is defined as the ideal of germs vanishing at p, which in general is not coherent.
Explanation: For simplicity consider an affine scheme, Spec A, with a maximal ideal p in A. I think here m_p should mean p, but as in the definition according to the index, m_p means pA_p, the maximal ideal in the germ.
[Thanks, I now understand your point! What I should have said was
, the ideal sheaf of 
. I hope this solves the problem. -R.]
(Previously posted)21.6.5: Here what does K(X’) mean? We have not shown X’ is integral so far. Also in the definition of X’ only the minimal polynomials for x_i is used, without using relations between different x_i. How is this enough?
Further issue: Now the K(X’) notation is removed, but there’s probably still issue concerning with relations between different x_i ‘s remaining. For example, take A=R, B=C, x_1=x_2=i. The problem is the relation x_1=x_2 is not captured by minimal polynomial of any x_i alone.
[Yes, that proof was botched. A patched version here here. The proof is from a discussion with Hikari Iwasaki. – R.]
(Previously posted)Also I met difficulties with 16.6.G: I could not prove S. -> \Gamma.\tilde{S.} defines a morphism on all of Proj \Gamma.\tilde{S.}, nor could I prove \Gamma.\tilde{S.} is finitely generated in degree 1. Could I get some help on these statements?
[By this I think you meant 15.6.G, but let me know if I am misunderstanding. I don’t think you actually need to prove that \Gamma. \tilde{S.} is finitely generated in degree 1! Part of the point here is that on the “geometry” side things are easier, while you have to work harder on the algebra side. The secret fact is that we’ll never need to understand the algebra side (at least for any of the fancy things we’ll do in algebraic geometry). So instead of trying to show an isomorphism of graded rings, try to show isomorphisms over “projective distinguished open sets”. After doing this problem, I recommend skipping all the later sections on the reverse map, and the saturation map. They feel like they should be useful, important, and not too hard. In fact they are none of these three. -R.]
Further question: This seems to only define an isomorphism of an open subscheme of Proj \Gamma.\tilde{S.} to Proj S. , as it is for an arbitrary map of graded rings to induce. I saw that it is an isomorphism on projective distinguished open sets, but I met difficulties proving they cover all of the newly constructed scheme.
[If I understand you correctly, you are basically done. The key thing is that have described a map from the newly constructed scheme to the old Proj; let me call it
. Furthermore, for any distinguished open subset of the old Proj, 
is an isomorphism above that distinguished open subset of the old Proj. But because the distinguished open subsets of the old Proj cover the old Proj, you are done. You have necessarily covered the new scheme! The key thing is that you have this map 
, which is global. The only thing you might be worried about is that in the language of Essential Exercise 7.4.A, 
, so that’s all you need to check. -R.]
Also to some previously posted issues no remarks were given. Are you going to give responses later, or are those issues malformed?
[I’ll post a response to that earlier post of yours, and to this one, when my in-line responses like this are complete. They are still in progress as of the time I write this. I deal with random ones as I find time. – R.]
18.2.3: ‘But the ith cohomology of the top row is precisely … except at step 0’: I think it should be ‘except at step 0 and 1’, since the complex on the top row is the augmented cech complex, and is exact at step 1.
[I see what you mean. In order to fix this without giving more cause for confusion, I now say “except at the start”. The reader thinking through the meaning of this sentence won’t get confused. -R.]
23.3.3: I’ve met problem applying the hint on inductively constructing the projective resolution of a complex from the bottom right: for example, consider only the bottom row of projectives surjecting to the complex
[My attempt to draw your equation array:
with downarrows
-R.]
? -> Z -> Z -> 0
| | |
v v v
Z -> Z -> 0 -> 0
Suppose I have decided everything to the bottom right of the ? position, where the bottom row is the original complex and the top row is P_{*,0}, and maps Z to Z are identities. Then the construction could not be proceeded to the ? item, since
?
|
v
Z -> Z should give a surjection, but
? -> Z
|
v
Z gives 0.
Are there additional induction hypotheses needed to make the hint work?
[See my reply below where I answer this, with a link to a revision. -R.]
23.3.8: Here around ‘G and Z^p commute’, an isomorphism between GZ and ZG is shown. But these two objects come with their own defining maps to other objects, since in category theory both ker and coker are defined to be diagrams, not only objects. I often find people claiming the importance of arrows when introducing Category Theory, but when people are acturally using categorical theorems and constructions, the arrows or naturality conditions are often ignored. Often it’s merely routine to verify these conditions, but there are so many of them and verifying them can be labour-intensive. What’s your attitude to such categorical work? Are readers supposed to verify the conditions themselves always even if it is omitted in the book, or is it just the norm for people to safely ignore such conditions when working? I’m curious to see your opinion.
[I think readers should verify the conditions themselves anyway. With some practice, it becomes a bit faster, but never that easy. But also you develop some sense of how far you have to go before you are sure that you see fully how it is going to work out. It is very much worth doing this because many times these things don’t work out the way the author thinks they do. Over time you get some sense that you can trust some authors on these issues, and distrust others, and you get the urge to be the kind of person that people can trust, which means having practice in actually verifying things. Frankly I worry that there are some parts of mathematics where the culture of “not worrying about it” has developed to an extent that the canonical literature has some amount of rot in it and that eventually it might collapse under its own weight. -R.]
23.5.4: ‘consider now our lift of a’ to our original class … to …’: This seems not quite a lift.
[Fair enough, although I hope no reader is confused. -R.]
Also naming conventions ‘Rightward filtration’ and ‘Rightward orientation’ are mixed. According to previous definitions, rightward orientation seems to use upward filtration.
[Good point. I am now careful to use “orientation” rather than “filtration”. -R.]
24.2.J: ‘If X and Y’ are k-schemes’: I think here it should be better rephrased that Y=Spec k
[Oh yes, you are right. I don’t know what I was thinking. Now edited. -R.]
24.3.O: Here I think ‘exact at the first step’ should be replaced with ‘exact except at the first step’.
[Agreed, fixed. Also grammar fixed in that sentence. – R.]
24.3.13: In the ‘similarly’ scenario, Y’ might be not flat over Z. For example, consider the case where pi = id_Z, and Y is a closed point of Z.
[Here you meant 24.4.13 presumably. Thanks, I’ve now added that hypothesis, that Y is flat over
. -R.]
24.5.6: ‘and Y has pure dimension n (all maximal chains of closed subsets have length n)’: ‘irreducible’ seems missing. Also this seems not the definition for pure dimension, as pure dimension schemes need not be catenary.
[You are right. I’ve just removed the parenthetical “(all maximal chains of closed subsets have length n)”. I think the exercise works now, but if I am wrong, please let me know!]
24.7.7: Here the condition provided in the parentheses (as M is flat over B) seems wrong.
[Here you recommend it should say “F is flat over B” if I’m not mistaken. Now fixed; I agree. -R.]
24.8.B: Here the hypotheses that C has no embedded points and curves being integral is missing.
[The parenthetical comment was in quotes in order to be informal and not quite correct. But your suggestion made me want to make it more correct, so it is now: “the degree of a finite morphism from a curve with no embedded points to an irreducible regular curve is constant”. -R.]
25.1.G: The ‘if you are feeling ambitious’ statement seems problematic, since the finitely presented version of the base-change theorem is not yet stated. Also in (a), the reducedness condition is not provided. It seems pretty impossible to me to overcome this lack of reducedness.
[I’ve removed the “if you are feeling ambitious” comment because I don’t want to distract readers with fancy abstraction. (I still might offer it to readers to think about it at a later date though.) For part (a): the cohomology and base change theorems don’t have any reducedness hypotheses! -R.]
25.1.J: Here it seems that M doesn’t have to be of finite rank.
[I could believe it but have to think about it a bit – do you use Nakayama at some point? I could also remark that Noetherian hypotheses could be dropped, although I should again be a bit careful before saying that without writing out the argument. -R.]
10.3.H: In the hint where X is the union of two affine open sets, is their intersection supposed to be distinguished in both of them?
[Of course, any route you take to get to the solution works. This is, however, a hard exercise! This is why it is in a double-starred section. Hopefully it is clear how important this is, because it makes it easy to generalize from the Noetherian situation to the finitely presented situation. -R.]
25.3: In the introduction of the Hilbert functor, is it needed to add flatness to the statement?
[Yes, thanks, fixed. -R.]
Thanks again for writing this wonderful book and make it freely available! After becoming busy in the following years, will you still fixing issues posted by comments? Or will you make some version final despite potentially having issues unsolved? Or will you post the latex source somewhere so people can submit pull requests? Thanks!
[I might hope to fix issues mentioned in comments, and imagine that I will fix at least some, and hopefully the ones that are mathematically or pedagogically the most serious or substantive. This particular site will at least be a public place for people to list known or potential issues, even if they never get corrected. Very possibly over time others will confirm or deny whether the issues are important. There will be a version that will be potentially final (and certainly an “official release”) despite having unsolved issues as well, and it may end up being truly final if I end up having essentially no time to do anything further. I won’t post latex source. I think a reasonable expectation is that there will be an “official release” (suitable for citing because it will be in a traditional format); a list of “official errata” here; discussion here of issues and errata which I will take part in as I find time; and of course discussion elsewhere as people ask questions of each other on other sites. Where I personally am on this spectrum: I try, but I also have finite time. So I realize that I will make mistakes, but I try hard not to, and try to stay in practice with checking some details. -R.]
February 9, 2024 at 11:32 am
Also could you leave a reply after replying by modifying the comment so I get a notification? Thanks!
February 10, 2024 at 11:01 am
Will do, although I’ll do so only when my response is complete enough to be worth alerting you. There is one earlier comment of yours where I’ve made a number of responses, but have a number still to go, so I haven’t left a reply just yet.
February 11, 2024 at 2:13 am
Hi! With 16.6.G: I think $V(\phi(S_+)) = \emptyset$ is actually the hard part people are struggling with, could you provide some help? Thanks!
February 15, 2024 at 7:20 pm
I’ve now answered everything in your comment. This reply here is a response to one of them, about 23.3.3. You are right that the previous exposition had a problem. Here is a revised version that should work (but everyone should let me know if there is still a flaw):
Click to access p665-6672024-02-11.pdf
February 17, 2024 at 11:25 am
Hi, professor Vakil! Have you seen this comment? It seems unanswered.
Hi! With 16.6.G: I think $V(\phi(S_+)) = \emptyset$ is actually the hard part people are struggling with, could you provide some help? Thanks!
February 17, 2024 at 5:43 pm
Here is a hint (for what is now 15.7.G, in the current version): There is an exercise called “A variation of the Qcqs lemma” (15.4.N in the version as of today, which I’ve also posted in my usual directory, although I’ve not made a post). The exercise is called “Important Exercise (to be used repeatedly)”. I think this makes the exercise you ask about tractable. Very possibly I should include this as a “possible hint” (especially if it is useful for people).
February 21, 2024 at 10:36 am
Hi, professor Vakil! Thanks for your help. However, I have been familiar with this lemma from the first place. I already know that for s considered as a section of either S. or Gamma. S.~, isomorphic affine patch is obtained that glues nicely. However, I could not prove these affine patches cover all of Proj Gamma. S.~ , that is, the elements of the form s generate the positive-degree ideal in Gamma. S.~ May I get some help towards this part of the proof? Thanks! I couldn’t work it out even though I already know the hint you just gave.
February 24, 2024 at 9:37 am
This request is now bumped forward to here.
February 11, 2024 at 3:28 pm
Thank you for the update, and all the hard work you’ve put in to this project. If you have taught or will teach grad algebraic geometry any time soon it would be great if you could post the exercises you’re assigning as homework — just doing the important exercises seems too few and doing all of them too much for a first pass through the material.
February 15, 2024 at 7:32 pm
I’m actually teaching from it now! If I get a chance, I’ll put some sample problems here. But I can answer your question more directly: I think if you do (or strongly attempt) all the important exercises, and seriously tackle a sampling of other exercises that catch your attention (and in particular those that you feel it is *necessary* for you to do), then that would be perfect for a first pass. You will hopefully find yourself coming back to do some more later on as you find them important. You should intend to skip the starred and double-starred exercises, although sometimes you will choose not to. Does that help? Let me know if I can say more.
February 17, 2024 at 12:33 pm
Right, that makes sense, and is sort of what I’ve been doing, using your old course page as a guide.
What I did when I organized a reading group out of the December 2022 version last spring was to match up the homework exercises assigned in the 2017-2018 course from the 2017 version to those in the newer version and do those. But given how much things have changed, it might be nice to an updated version if time and circumstances permit.
February 20, 2024 at 11:29 am
Don’t know if this helps, but Aaron Pixton taught from Ravi’s book at Michigan last term, and I am doing it now; our assignments are visible at https://public.websites.umich.edu/~pixton/631/ and https://dept.math.lsa.umich.edu/~speyer/632/ .
February 20, 2024 at 12:58 pm
This is great, thank you very much.
February 21, 2024 at 5:37 pm
Thanks for posting these links, David, I think they are even more helpful than something I would say myself!
February 13, 2024 at 10:30 am
Nit: p. 147, 4.4.11, line 7: “those prime ideals” => “those prime ideals *of ℤ*”, perhaps?
February 15, 2024 at 7:29 pm
Sounds reasonable, now done!
February 13, 2024 at 12:48 pm
Hi Ravi. Typo nit here: In the parenthetical part of 2.7.8, ‘… if a sheaf has value in…’ –> ‘… if a sheaf has *values* in…’. But it was there in the previous version.
February 15, 2024 at 7:27 pm
Thanks, fixed!
February 15, 2024 at 10:37 am
I don’t know if you want suggestions for further exercises, but to me, it seems like Exercise 15.4.Y would go naturally with a companion exercise 15.4.Z:
Let X, L, s, X_s and F be as in Exercise 15.4.Y. Let t be a section of F on X such that the restriction of t to X_s is 0. Then there is a positive integer n such that s^n \otimes t is 0 as a section of L^n \otimes F.
February 15, 2024 at 10:43 am
Or perhaps just split 15.4.Y into two parts. Part (1) Any section over X_s can be extended to a section over X after multiplying by some s^n. Part (2) If we have two such extensions, they become equal after multiplying by appropriate higher powers of s.
February 17, 2024 at 9:19 am
How is this? (it is 15.4.M in this version)
Click to access p446-4472024-02-16.pdf
February 17, 2024 at 12:09 pm
This conveys the point I was making, thanks! I think I would have written it differently, but I’m not confident that mine would have been better.
February 16, 2024 at 8:32 pm
On p. 151 you define *finitely generated graded ring over A*, and then in Ex. 4.5.G you introduce the notion of a *finitely generated graded A-algebra*. 4.5.G asks the reader to show that these are equivalent, which is fine, but it’s worth noting that they are both equivalent to the condition that S_• is a finitely-generated A-algebra. The latter is a notion we are already familiar with, so it’s not clear why there’s a need to introduce two new terms when they just mean the same as one we already have.
February 17, 2024 at 8:22 am
The main reason is that in 4.5.G(a), it says generated over A = S0 by a finite number of *homogeneous* elements. If I just said “finitely generated A-algebra”, then I’d need to additionally ask the reader to show that we can take the generators to be homogeneous, which adds an additional complexity to the problem.
February 17, 2024 at 9:54 am
Fair enough, but the fact that we can take the generators to be homogeneous is a fairly trivial fact about homogeneous ideals once we have 4.5.F(a).
Perhaps what I stumbled over is the fact that the terms “finitely generated graded ring” and “finitely generated graded algebra” are very similar, but the definitions are very different, which is all the more confusing since ultimately they mean the same thing.
I think the meat of the exercise is the parenthetical comment, which, incidentally, goes both ways – any homogeneous subset of S₊ generates S₊ as an ideal iff it generates S_• as an A-algebra.
In general I agree that the *facts* that are being described here are important and useful, but I don’t think the terminology is very helpful in aiding understanding or remembering them.
February 17, 2024 at 3:47 pm
Nit: on p. 52, sentence before 4.5.H: “The points of Proj S are the set of homogeneous prime ideals…” – maybe this should just be “The points of Proj S are the homogeneous prime ideals…”
February 17, 2024 at 4:45 pm
Agreed, done, thanks!
February 18, 2024 at 9:53 am
On p. 153, 4.5.8, you define V(T) for a set T of homogeneous elements of S_*, and at the end of the paragraph you explain that you deliberately stick to this narrow definition (as opposed to defining V(T) for arbitrary subsets of S_*). Then in 4.5.K(b) you ask the reader to define I(Z), and show that it is a homogeneous ideal. Finally, in 4.5.K(c) you ask the reader to show that V(I(Z)) is the closure of Z, but this now requires V() to be defined on subsets that are not composed solely of homogeneous elements.
I suspect you *do* want to define V(T) for more general subsets T; I expect the Galois connection properties of V and I will be used repeatedly, even if you don’t use that terminology.
February 21, 2024 at 5:34 pm
Actually, although you absolutely could, as you observe, and it really isn’t a problem, as you implicitly observe, I explicitly avoid the issue, because a nontrivial number of readers ended up spending a lot of time on the issue of graded vs. nongraded, and I didn’t want them to be distracted. This is again one of those this situations where rationally it looks like it makes sense to do it one way, but when humans are involved as readers, it seems better empirically to do it a different way….
February 19, 2024 at 11:35 am
Potential typo: in exercise 15.1.D, should the structure sheaf on A^{n+1}\{0} have \{0} in the subscript?
February 19, 2024 at 1:03 pm
I think this is not a typo: first form the affine space, then excluding one point from it.
February 21, 2024 at 5:34 pm
I realize the two of you are looking at different versions — this error was there in a recent version, and is now fixed in the most recent version. (I haven’t checked if it is in the most recent public version, but at least it is fixed in the version to be posted soon.)
February 19, 2024 at 6:21 pm
Dear Ravi,
Thank you so much for the revisions! A few comments, which might just be simple confusions. Does Exercise 6.6.X require a Noetherian hypothesis on A for part (a) and a locally Noetherian hypothesis on X in part (b)? For a possible counterexample to part (a), consider A = k[x_1, x_2, …]/((x_1)^2, (x_2)^2, …). Then A is supported at the single point of Spec A and also finitely-generated over itself, but not of finite-length (e.g. the infinite ascending chain of strict inclusions (x_1) \subset (x_1, x_2) \subset …). If this exercise does require the Noetherian hypotheses, then the advertisement of a converse in Exercises 6.5.K and 6.5.M might also need to be edited.
Kindly,
Daniel
February 19, 2024 at 10:55 pm
Also, I think the text in 6.6.X(b) should be “supported at finitely many points of X” as opposed to “supported at finitely many points of Spec A”.
February 21, 2024 at 5:33 pm
Thanks, fixed!
February 20, 2024 at 12:11 am
Another comment (sorry for not making all these in a single message!): should there be a “locally-finite k-bar scheme” hypothesis on X in 6.6.Y(b)? Otherwise, I think the composition series for each stalk O_{X, p_i} across the discrete points p_i \in X might not given a composition series for O_{X, p_i} as a k-bar vector space, so the length of X there might not correspond to the dimension, as intended.
February 20, 2024 at 12:24 am
Correction: the suggested hypothesis should be “locally finite-type k-bar scheme”, not “locally finite k-bar scheme”
February 21, 2024 at 5:33 pm
Yes, you’re right, thanks!
February 21, 2024 at 5:32 pm
Good point!
Quick check (because I’m typing this late at night): does adding Noetherian hypotheses fix both issues? More important: could you solve the problem as you stated it?
February 21, 2024 at 6:06 pm
Yes to both!
February 21, 2024 at 6:07 pm
Thanks, perfect, I’ve made the changes!
February 20, 2024 at 11:23 am
Exercise suggestion: Definition 11.3.2 has the parenthetical remark “once you show that a locally closed em-
bedding whose image is closed is actually a closed embedding”. Maybe make that an actual exercise in Section 9.3 and reference it here?
PS I appreciate you explicitly saying that “diagonal is closed” implies “diagonal map is closed embedding”; I wasn’t sure, and was rereading this section exactly to look for an answer, so thanks!
February 20, 2024 at 11:38 am
Oh, another (easy) exercise suggestion. Exercise 11.3.M almost, but doesn’t quite, say the following: Let X–>Spec A be a scheme over an affine base. Then X–>Spec A is separated if and only if, for all affine opens U and V in X, (1) the intersection U \cap V is affine and (2) the map O(U) \otimes_A O(V) –> O(U \cap V) is surjective. Moreover, it is enough to check that this holds as U and V range over the sets in any affine cover X = \bigcup U_i.
Maybe this should be Exercise 11.3.N?
Let me know if the exercise suggestions are annoying. I do know you have to stop adding exercises somewhere.
February 21, 2024 at 5:40 pm
I thought about this. We never use it. But it is indeed nice. I remember seeing it in EGA and thinking “this is so easy – why had no one pointed this out to me before”? So it is actually what I used to prove Prop 11.3.8 shortly before the segment you are discussing, showing that projective space is separated (Approach 1)! (I actually don’t immediately see that Exercise 11.3.M almost shows the statement you say.) So now I’ve added it; also clearly state that we’re not going to use it (so people can skip it) but that they may like it (so people might not skip it). I mention that it is a partial converse to the “affine intersect affine = affine” (actually it says what in addition you need for separatedness to hold of course), and then point out in the problem that we actually did this argument earlier. See here:
Click to access davidspeyer-p321-2024-02-21.pdf
February 21, 2024 at 4:28 am
Exercise 17.1.H. could in my eyes do with some clarification. Precisely: You want us to prove that the map \pi is the affine map that we get from the total space (i.e. the line bundle) associated with O(-1) on P^n. To me the suggested path in the hint seems to a priori assume that the sheaf of algebras (say R) inducing \pi is indeed the sheaf underlying a total space (i.e. R \cong Sym(L^\vee) for some locally free sheaf L on P^n).
In my opinion, one also has to prove that this is truly the case.
It might, however, also be that I have been looking for too long at the exercise and now I am just confused …
February 21, 2024 at 5:32 pm
I think you are overthinking it. Or at least, I would not want to enforce on the reader the kind of thinking you are doing. My suspicion is that you know more than most at this point, and hence you are giving a more powerful solution, but at a cost you clearly appreciate. Let me tell you my thoughts, and then what people have done. For me, this is a somewhat open-ended question: given what you know, how would you tackle such a question? In particular, it is not the kind of question for which there is one particular solution which is intended to either enlighten you, or give you practice. Instead, you have to think things through in a more messy way, which helps for the development of a better more robust understanding. One common kind of solution is to try recognize what is being described as something which is the total space of a line bundle, in that it is locally trivial over each of the standard open sets. Then you figure out what the transition functions are between them (thereby confronting a possible sign issue, where you might get as an answer O(1) rather than O(-1)). Then you put the pieces together to finish the problem.
February 23, 2024 at 2:57 am
Thank you for clarifying. I will at some point have to think it through once more to see if it makes sense now.
February 21, 2024 at 7:06 pm
In 4.5.K you ask the reader to “define I(Z) ⊂ S₊”. It’s fairly obvious that I(Z) “should” just be the intersection of all the elements of Z, mimicking the affine case, but that needn’t be contained in S₊. So presumably one needs to also intersect with S₊.
“Inserting” S₊ in this way seems a little inelegant. Maybe it’s the solution you’re looking for though. Or maybe there’s another way to define I(Z) that naturally results in something contained in S₊. In either case, I think you should provide a bit more guidance, either explaining why the condition I(Z) ⊂ S₊ is desirable, or else giving a clue how that would arise naturally.
February 24, 2024 at 9:35 am
That was a subtle point, and I was hoping no one would notice that!
I want to develop all the discussion in terms of ideals contained in the irrelevant ideal, and everything works then. In particular, we can apply part (a). But I don’t want to tell people everything that can go wrong if we made different choices, because there are so many different such stories that could be told. Worse still, sometimes not too much goes wrong, but I don’t want to get into all the details of these side issues and the pros and cons of small changes, and instead I want to hurry them along into a later chapter.
If I point out this issue here, then a number of readers who would *not* have slowed down here will then slow down here, so that’s why I’m passing by this point without comment in the notes. A super-tough thing with this entire project is that it that I find it very hard to get people through all this material over the course of a single year without curating and making some arbitrary choices without going into detail about why I am making those choices.
(I have noticed that a number of points where I prefer the reader to not read something, but I want to have it there as a reference. And I found out that it instead ended up slowing people down because they couldn’t help reading it and asking themselves questions about it.)
February 24, 2024 at 4:45 pm
Alright, so I just want to confirm then, that the intended solution is to define I(Z) to be the intersection of the ideals in Z, together with S₊, in which case we don’t get the Galois connection that simplifies so many arguments.
Or is the intent instead that we should also restrict our attention to prime ideals contained in S+?
February 22, 2024 at 10:26 am
Hi, professor Vakil! Thank you very much again for the great book! Here are some more issues I’ve come up with after the previous batch:
26.1.4: Here M needs to be nonzero, otherwise (i) and (ii) will be true for any n but (iii) could not be true for any nonzero n.
[Good point, now fixed! – R.]
21.6.E: Here do we have the condition that X and Y are smooth? It isn’t mentioned in the statement, but appears in the hint.
[Right! Here is a further hint: they are “mostly smooth”. – R]
13.4.2: Bertini’s Theorem stated here seems not general enough for subsequent usages. I think a sufficiently strong version should cover the case of a smooth subvariety of P^n_k of pure dimension d, intersection m hypersurfaces whose degrees are fixed respectively, and still have an open locus in the parameter space with smooth intersection of pure dimension d-m.
[Response still to come. -R.]
20.1.C.(b): What does generally choosing mean here? In the usual sense, i.e., an open locus in the total parameter space, we will in fact have a regular sequence thanks to Cohen-Macaulayness. However, not having C-M-ness at this stage, a less general approach is to choose these hypersurfaces one by one, each arbitrarily in an open locus in its own parameter space determined by the previous ones. This will be less general because open subspaces in fibers in a fiber bundle don’t necessarily give an open subspace of the fiber bundle, however it will be more doable to readers at this stage. I hope there could be some clarifications here.
[Response still to come. -R.]
18.4.L. How could one write L into the form of O(sum n_j p_j) where are p_j couldn’t lie in a specific finite set of points? After counselling one strong friend of mine, I got this approach:
a) The target is to construct a rational section invertible at each p_j.
b) Divide L into quotient of two very ample line bundles. WLOG we may assume L is very ample.
c) If k is infinite, we may choose a suitable global section not vanishing near these points.
However, this approach requires the hypotheses that k is infinite. Even if we could base-change to infinite k in first place while preserving degree, since it is unmentioned in the hint, I don’t think it is the intended solution. What’s the intended way to do this?
[Here is a hint. If you have finitely many points in projective space over a finite field, you are right that there need not be a hyperplane missing them all. But (you can show that) there is a positive degree hypersurface missing them all! -R]
20.1.F. In the index, there are so many items for definition of effective Cartier divisors. Hopefully there could be some hint in the index of what kind of definition these links are pointing to.
[There is only one entry for the definition, which is in bold. They are mentioned many times, and are currently indexed when they are mentioned. Later perhaps that will be trimmed down. But certainly the index won’t get any more detailed, as it is already too long. There are going to be some rather severe trade-offs to be made in making the final index unfortunately, as there are starkly competing priorities which cannot be reconciled. -R]
20.2.3. There’s some vagueness in the definition of C and D here: They are defined to be effective divisors, which might be considered Weil divisors by default. Then it’s claimed that C and D couldn’t have embedded points, which is plainly true as Weil divisors.
[Yes, you can see the deliberate ambiguity here, because there should never be any confusion about why we want both meanings. “Embedded point” only applies to schemes, so it can only apply to their incarnation as effective Cartier divisors. It is not defined for Weil divisors. I have added a bit more text though, to clarify this point, although any addition of text also obscures at the same time. – R.]
20.1.I. Here in the sum containing q(m+i+j), m+i always appears as a whole. Maybe they are duplicated? I think only getting rid of one of them will be fine.
[Thanks – they were indeed duplicated. I’ve deleted “+i”, and changed the “i” in the next sentence to “m”. – R.]
20.2.E. There’s a more direct hint: consider square of elements in Pic. One gives all integers and another gives all even integers. Maybe the current hint prevailing the general case better, but it is frustrating to spend decent time finding out what line bundles are effective after proving the conclusion in another way.
[That’s very nice! I now added: “A simpler approach but without the same generalizations: show that the self-intersections of all the elements of $\Pic$ give $\Z$ in one case and $2 \Z$ in the other.” Why still give the harder approach first? Because I want to mention the generalizations, which lead into some important ideas. But I also don’t say that the approach you mention, interpreted appropriately, also is part of an important theme in geometry: an invariant of this surface coming from the nature of the intersection form. -R.]
20.2.I. Here Q and I are sheaves on X, not sigma(X), although they are canonically identified.
[True, but I think no one has gotten genuinely confused by this. This was a case where there was a choice between precision and clarity. -R.]
The hint uses transition matrixes, which seems cumbersome. I think here operating on Sym^. constructions would be cleaner.
[I actually think your approach is better, and the philosophically right approach. But it requires more maturity to see, and so the approach I tend to point out to those seeing it for the first time is the sort of thing that they might think about by “following their nose”. They will then later appreciate why the right point of view is very helpful. And if they immediately give the argument you prefer, all the better — I strongly do not want to direct readers to solve exercises in one particular way. I have found myself often changing “hint” to “possible hint” for precisely this reason. -R.]
20.2.L. Here it is stated that I will develop ‘local charts’ for F_n. However, by identifying Pic with Cl and excising E, I think this result could be shown almost immediately. What’s the intended approach?
[As with my previous comment, I deliberately have no “intended” approach. I even prefer it if the students don’t always take the fastest shortest approach, and instead have first arguments that are complete but inefficient, which involve not just understanding the shortest road between two ideas, but also some of the landscape nearby. I think your approach is better but more sophisticated, and I am happy if they solve the problem by any means necessary, and move on. They will then acquire that sophistication subconsciously over time, and shouldn’t worry about getting it now. I also hope that people aren’t afraid of working with local coordinates if they need to. -R]
17.2.4. There’s a typo here: ‘whether PF should be defined _as_ … or …’ should be ‘whether PF should be defined _with_ … or …’
[If I understand your point correctly, what I wrote is fine. I think it is a question of linguistic convention. -R.]
20.2.10. Here in the definition of Hirzebruch surfaces, P should be used instead of Proj, which works with quasicoherent graded algebras.
[Thanks for catching that! Now fixed. -R.]
20.2.O. Here I think maybe ‘namely C’ should be clarified about the way C is identified with the rational function 1. Also, why are readers suggested to write down sections using local charts for F_n? Doesn’t it follow directly from 20.2.M?
[Oh yes, that is cleaner! Now changed. -R.]
20.2.18. Here I think the coeffecient 2 is missing for n L \cdot R in the expression of L’ \cdot L’.
[Thanks, fixed! – R.]
20.2.19. Here I think the application of the Hodge Index Theorem for the case where
, but 
. I think the trick for 
case, since the theorem's conclusion is symmetric for 
and 
. Similar reasoning might be invalid in multiple places in the proof.
[I don’t understand what you are trying to say here. (I’m not saying you are wrong; I just don’t understand what the statement is, or what sentence in the notes it applies to, or how. – R.]
Also I’ve posted a reply asking for further aid about 15.7.G. I saw you replied to almost all other comments, so did you miss this one, or was it ignored on some purpose? I’ll repost the content here:
Hi, professor Vakil! Thanks for your help. However, I have been familiar with this lemma from the first place. I already know that for s considered as a section of either S. or Gamma. S.~, isomorphic affine patch is obtained that glues nicely. However, I could not prove these affine patches cover all of Proj Gamma. S.~ , that is, the elements of the form s generate the positive-degree ideal in Gamma. S.~ May I get some help towards this part of the proof? Thanks! I couldn’t work it out even though I already know the hint you just gave.
[Response still to come. -R.]
Thanks again for your inspiring book and huge effort answering comments!
February 24, 2024 at 10:21 pm
Hi, Professor Vakil! Thank you very much for helping! Here are some further clarification requests to your response:
21.6.E: Here do we have the condition that X and Y are smooth? It isn’t mentioned in the statement, but appears in the hint.
[Right! Here is a further hint: they are “mostly smooth”. – R]
[[Is it true that in this exercise X and Y are smooth? If so, why do we need the hint? If not, what does ‘Right!’ mean?]]
18.4.L. How could one write L into the form of O(sum n_j p_j) where are p_j couldn’t lie in a specific finite set of points? After counselling one strong friend of mine, I got this approach:
a) The target is to construct a rational section invertible at each p_j.
b) Divide L into quotient of two very ample line bundles. WLOG we may assume L is very ample.
c) If k is infinite, we may choose a suitable global section not vanishing near these points.
However, this approach requires the hypotheses that k is infinite. Even if we could base-change to infinite k in first place while preserving degree, since it is unmentioned in the hint, I don’t think it is the intended solution. What’s the intended way to do this?
[Here is a hint. If you have finitely many points in projective space over a finite field, you are right that there need not be a hyperplane missing them all. But (you can show that) there is a positive degree hypersurface missing them all! -R]
[[I’ve thought about this idea previously. However, positive degree hypersurfaces don’t give rational sections on the line bundle we care, which is assumed to be O(1). I met serious obstacles trying to overcome this. Moreover, I think a hint might should be better added: There has been many pages since the idea of factoring a linear bundle as quotient of very ample bundles and this exercise. And the idea of factoring as quotient of very ample bundles hasn’t got much usage, so readers might forget this is a thing at this stage. It happened to me and I had to seek help from a talented peer student. ]]
February 25, 2024 at 9:54 am
Hi, Professor Vakil! Thank you very much for helping! Here’s further explanation for question about 20.2.19:
Consider the ‘If A * L <= 0, then we are done without even using H'. Can this be understood that if A * L < 0, then without using any conditions about H, we will have the result that L * L <= 0? But this is not true, for example take L = -A .
February 23, 2024 at 12:27 pm
My students raised a subtle point about the discussion in 15.4.7-15.4.K . After talking to them a while, it looks to me like this is exactly what 15.4.8, 15.4.J and the parenthetical Caution in 15.4.K were meant to address, so maybe everything is already in the text, it just somehow didn’t come together.
Suppose that R is a Noetherian R1 domain (but not yet normal). Let D be a formal, nonnegative, sum of codimension 1 primes. Then there are two different possible meanings of “D is principal”:
(1) There is an element f in R such that div(f)=R
(2) The ideal O(-D) in R is principal.
We have (2) –> (1), but not vice versa. To see this, consider the pinched plane R = k[x^2, x^3, y, xy] and let P be the prime ideal . Then div(y) = [P], but the prime ideal P itself is not principal.
If we impose that R is normal, then these are equivalent, and I guess this is the point of important exercise 15.4.J (which perhaps I erred in not assigning).
****
Where my students were confused is that 15.4.K cautions the students to distinguish between two meanings of [locally] principal and, after some thought, my students came up with (1) and (2). Defn 15.4.7 is definitely using (2). But the Hint for this problem seems to be pushing towards definition (1).
Is the point just that they are supposed to explicitly say “This ring is normal, so Exercise 15.4.J says that (1) and (2) are equivalent, and now we can use the Hint.”?
February 28, 2024 at 6:58 am
Oops, I jut noticed that the definition of P didn’t post, probably because of angle brackets. P is the ideal generated by y and xy.
February 24, 2024 at 9:29 pm
Dear Prof. Vakil,
Thank you so much for this excellent book!
I have a kind of confusion of the figure 4.9: ” x^2+y^{2}=z^{2} in P^{2}” actually points to the circle rather than the sphere, right?
Please forgive the simplicity of my question. I am just beginning to learn algebraic geometry.
February 25, 2024 at 8:36 am
Yes, that’s right. If that’s the way it looks, I’m happy.
I remade this figure, and I’m hoping you are looking at the current version, which is hand-drawn. If you are looking at an older version, please take a look here (e.g., at the February 21, 2024 version).
If the figure does not clearly indicate to you what I said, let me know too!
So to answer the mathematical point of your message: I visualize
as the sphere 
. (Actually, 
is basically this sphere, except with opposite points identified, so when looking at this cartoon, you should secretly think of the opposite points as being identified.) Or better, I prefer to think of it as a sphere with “infinite radius”; it is a sphere way out at infinity.
Then in
, I visualize the conic 
as the intersection of the cone (with the same equation) with the sphere (of infinite radius). (You might say “doesn’t this mean it is two circles?”, but as in the previous paragraph, I’m identifying opposite points of the sphere implicitly.)
I hope this helps!
February 25, 2024 at 10:48 am
Thanks! The version I had is Feb06, and the figure in the current version is perfectly clear.
And also thanks for your explanation! At first, I indeed did’t understand why you put a sphere in the figure, but now everything makes sense.