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April 2015 version

Posted by ravivakil under

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[35] Comments
My hope of a new version each month has clearly not worked out! But here is one for April, just before the month ends. As usual, a random selection of corrections and minor improvements have been dealt with, and many more remain to do.

The April version is posted at **the usual place** (the April 29, 2015 version).

May 4, 2015 at 11:54 pm

Hey Ravi, I’d like to give a suggestion, if possible. I very much like in Chapter 25 that you discuss what the ‘smooth manifold’ analogue of smooth, unramified, and étale is. It would be nice to give an exercise telling people that this analogy is precise in the complex setting (e.g. étale iff analytification is local biholomorphism).

Also, I think that it would be nice to give an example showing why the naive ‘smoothness’ is not the thing that we want–or, at least, not the thing that we have defined. Giving the example of a family where and the map is just projection to . This gives a very nice picture of a family of ellpitic curves degenerating to a cuspidal cubic. It is also a map which is holomorphic and proper, but is not smooth in the algebraic geometric sense (since its special fiber is singular). The problem then, of course, is that it’s not submersive at .

Just a thought.

December 1, 2015 at 11:00 am

Thanks as always for your very helpful comments!

In response to the first paragraph — that is a great idea, and I will add this. (Currently it is on the “to-do list”, meaning it will be done, but isn’t done yet.) About the second paragraph — I don’t fully get it — do you mean that one might naively think that every map from “smooth to smooth” is smooth?

December 3, 2015 at 5:07 pm

Hey Ravi,

No, what I more meant is that one might imagine that if and are smooth varieties over then a naive guess as to what it means for to be `smooth’ is that is holomorphic since holomorphic is stronger than (=smooth)—it’s the `smooth maps’ in the category of complex manifolds (i.e. the analogue of smooth maps in the category of real manifolds).

Of course, this is not correct, since, as you yourself point out in your text, `smooth’ is more like `submersion’ than actual smooth. In particular, there is no reason that a holomorphic map need have smooth (in the sense of complex analytic spaces, or complex varieties) fibers whereas smooth maps should satisfy this.

My point is just to emphasize that even though holomorphic maps are smooth they are more the analogue of just *maps* (regular maps) than smooth maps. The example in the second paragraph is just a nice geometric cartoon of such a situation (of course, the squaring map on $\mathbb{A}^1$ works perfectly well also).

Best,

Alex

December 3, 2015 at 5:11 pm

Sorry, I think the following summary might be more helpful:

You discuss in the smoothness chapter that `smooth map of varieties’ is not the same thing (intuition wise) as ‘smooth map of real manifolds’. I feel like it might be helpful, since this is closer to actually possibly being true, to emphasize the complex version. Namely, `smooth map of varieties’ is not the same thing (intuitively, or through analytification) as a ‘holomorphic map of complex manifolds’—in fact, the latter is correctly the analogue of just a regular map of varieties (i.e. a regular map), with literal equality in the projective case.

Thanks!

Best,

Alex

May 26, 2015 at 5:02 am

minor typo : page 130 first line “identityand” should be two words.

May 26, 2015 at 11:02 am

Thanks — now fixed! If you see any more like that, please let me know!

May 29, 2015 at 2:09 pm

2 minor typos: repetition in paragraph 3, page 203 “We say a \in A is integral over B if a satisfies some monic over B if a satisfies some monic over B if a satisfies some monic polynomial…” and a cross-reference tag error in exercise 8.1.D at the bottom of page 223.

September 29, 2015 at 11:08 am

Thanks, now (finally) fixed!

June 16, 2015 at 2:25 pm

Hi Ravi,

Do really need to use miracle flatness for the 27 lines? (I’m looking at the top of p. 691 of what I think/hope is the latest version.) You have proved that you have a projective morphism from one 19 dim’l variety to another which induces an injection on Zariski tangent spaces at closed points. The target is smooth, and hence the source must be smooth (consider dimensions) and the morphism must induce an *isomorphism* on Zariski tangent spaces at closed points, thus is a proper etale morphism, hence is finite etale. Probably for a finite etale morphism you can prove that the fibre dimension is locally constant in a more direct way than via miracle flatness. (I think it’s nice to squeeze as much as possible out of the tangent space computations, since it is really a strong application of Zariski tangent space ideas and their relationship to etaleness/smoothness.)

Cheers,

Matt

June 26, 2015 at 9:27 pm

Matt, I think there is an expository complication caused by the fact that all discussion of flatness aspects of completion only comes later in the notes. Let’s see how this becomes an issue when trying to push through the idea.

Firstly, we know that a map between smooth varieties which is an isomorphism between tangent spaces at a pair of rational points has induced map between the corresponding completed local rings necessarily an isomorphism: those are formal power series rings over the same residue field in the same number of variables, and the map between these completions is surjective (by successive approximation, building off of the tangent space isomorphism), so it is an isomorphism (by dimension considerations from commutative algebra — in what generality is the dimension theory of noetherian rings developed in this notes? — or by the more elementary but somewhat more algebraic fact that a surjective endomorphism of a noetherian ring is an isomorphism).

We would like to then say (without any appeal to flatness aspects of completion, which are only discussed later in the notes) that the fiber over any geometric closed point has completion at each closed point equal to the algebraically closed ground field (whence each such fiber scheme is finite etale), but this entails knowing that completion commutes with quotients by an ideal, which I don’t think is available yet at this place in the notes. Granting it is somehow deducible, the proper map of interest is therefore quasi-finite, hence finite. But unless one invokes later results on completions (such as that completion of the base for a finite map yields direct product of completions upstairs, or that completion is faithfully flat for a local noetherian ring) it isn’t clear how to bootstrap to relate fiber degree at various points (or how to get flatness).

I haven’t looked at how etaleness is defined or especially how its basic features are developed in these notes, but I am a bit doubtful that without invoking in some way the results on completion that only occur later one can push through this suggested argument through tangent spaces (but I concur that this alternative approach has somewhat more vivid geometric appeal; maybe it can be turned into an Exercise later, with forward-reference to it in the 27-lines discussion).

June 26, 2015 at 12:30 pm

I think there’s a typo in exercise 27.2D: it should be that either x_0 + \omega x_i = 0 and \omega is a cube root of 1, or that x_0 = \omega x_i and \omega is a cube root of -1.

Also, I’m a little confused about Exercise 27.2.A and exercises 27.2.E-27.2.G. They seem to be asking very similar if not the same thing, and the hints also seem to be very similar.

June 29, 2015 at 5:30 pm

Thanks! About your first point: whoops! Now fixed. About the second: I must have added them at different times — I have now removed 27.2.A. Thanks for catching that!

June 30, 2015 at 8:24 pm

Tiny little typo: exercise 2.5.A has F_x when it should be F_p, I think.

July 1, 2015 at 7:23 am

Thanks, now fixed! (I am surprised this one lasted so long…) If you see more, please let me know!

July 3, 2015 at 10:20 pm

I saw more! In particular another even tinier one with the hint in 3.2.G: multiplication by x is only an isomorphism if x isn’t 0.

October 17, 2015 at 5:56 pm

Thanks, now fixed — another one that I am surprised lasted so long!

July 3, 2015 at 11:01 am

Wanmin Liu had earlier shown me an example of these notes converted by pdf2thmlEX here. He wrote to me (on Oct. 9, 2013):

“Recently I’ve managed to produce a better version, which is

optimized for online publishing. Now the pages are splitted and stored

into separate files such that pages are only donwloaded upon user

requests. This is especially useful for users who just want a quick

review of a few chapters in the book. Of course the bookmarks are also

preserved and shown on the left side of the web page.

I’ve put the 1st chapter of the book on

http://www.cse.ust.hk/~luwang/out/FOAGjun1113public.html ”

Click here for a direct link.

This looks very cool!

July 19, 2015 at 9:26 am

minor typo: in 2.2.14 (page 78) symbol V is used both for the vector bundle and for the open subset in X.

September 17, 2015 at 11:12 am

Thanks, now fixed! (How did that one last so long?!)

July 22, 2015 at 1:30 pm

Do you prefer long lists or just spur-of-the-moment additions to the errata?

The proof of 12.4.2 seems a little off to me. The Jacobian matrix in 12.4.3 ignores ; it seems to me that we should use the projective tangent space from 12.3.J and note that the Jacobian has corank , and require that this not drop when we append the column with the .

July 23, 2015 at 5:53 am

I’m happy with either long lists of spur-of-the-moment errata! I will also look at your comment soon too…

October 17, 2015 at 5:57 pm

Follow-up: you are completely right of course — this entire section needs editing as well, so I have kept this with my to-do list for this section.

August 4, 2015 at 6:07 am

Hi Ravi,

I have some questions regarding sections 9.1.6 and 9.1.7 in your notes:

Firtsly: In your notes you define “open subfunctor” on page 248, and you do not define “subfunctor” at any point before this. I am wondering whether it follows from your definition of open subfunctor that an open subfunctor is also a subfunctor, or did you forget to add this (or is it even relevant ?).

I also notice that in your definition of (open) cover (of a presheaf), you do not require that the cover $\{h_i: \to h \}_i$ has the property that $\coprod h_i \to h$ is locally surjective.

Using your definitions I have solved exercises 9.1.E through 9.1.H, however I find it very hard to do 9.1.I without the two extra conditions I mentioned above. So I am wondering if you have forgotten to write something, or if what I mentioned follows from your definitions, or if it is needed at all to do exercise 9.1.I.

By the way I see similar definitions to yours on the stacks project (http://stacks.math.columbia.edu/tag/01JI ) , where the things I have pointed out are part of the definitions.

Thank you.

August 9, 2015 at 5:27 pm

I would like to add that I am a big fan of your book and I am only asking this to make things clear for myself and others who might be struggling with the same exercise/concept.

August 12, 2015 at 12:59 pm

For the first condition: is the problem injectivity? I think it follows from the definition given.

August 12, 2015 at 3:22 pm

Yes for the first it is indeed injectivity. Can you point me in the right direction of proving that?

Thanks.

August 12, 2015 at 4:19 pm

I think the right level of abstraction is this: say I have a map of sets and mapping to the same . If I want to conclude that then it would be enough to find a map with in its image such that the base change of by is injective. This is the situation we’re in! Remember that Yoneda tells you how to think about morphia a .

August 23, 2015 at 10:27 pm

* The proof of 20.1.4 should reference 16.6.F instead of 16.6.C.

* Speaking of 16.6, I think the meaning of “complete linear series” in 16.6.A will be confusing to some. We’re not working over a field and (apologies if wrong) at this point in the book it’s not clear that this thing is finite dimensional in any sense. In the end, and in the proof, all we need are *some* sections that do the job.

* Have you given any thought to including a (double starred?) section with suggestions for working with Cartier divisors when one inevitably sees those in the literature?

With much appreciation,

September 22, 2015 at 4:10 pm

Thanks Hoot!

About the first jot: Thanks now fixed!

About 16.6: Agreed — will be fixed!

About the third jot: do you have examples of facts that you would find worth seeing?

November 17, 2015 at 10:46 am

Regarding Cartier divisors: I’m not an expert so not really. I just imagine people running into these things in Fulton, Kollár-Mori, Lazarsfeld, etc. and suddenly having to deal with this weird sheaf $\mathscr K$. It seems like you’re really close to having a definition in the (locally?) noetherian case with these pairs of a bundle an a “meromorphic” section. I don’t know. Maybe it’s not worth doing when all other books are going to use another definition anyway?

December 29, 2015 at 5:18 pm

Do you mean something along the lines of Matt Emerton’s excellent stack exchange answer here: http://math.stackexchange.com/a/1943/45028 ?

January 3, 2016 at 8:50 am

Of course one has to worry about bloat but I think putting something like this double-starred at the end of, say, 14.3 would be amazing.

January 3, 2016 at 9:02 am

So what Matt Emerton said (which I think he said particularly concisely and clearly) is the information you were looking for? (This could be in an exercise, suitably hinted.)

September 17, 2015 at 11:31 pm

In example 7.3.9 the command for the index is a bit off resulting in some garbled text.

September 22, 2015 at 4:05 pm

Thanks Pieter — now fixed!