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November 2015 version

Posted by ravivakil under

Actual notes
[18] Comments
Following my goal of posting current versions every month, the November 28, 2015 version is in **the usual place**. There are many small changes, but nothing that should particularly make you want to download it.

As always, I have a big list of emails, responses, etc. that I want to respond to, and a number I intend to respond to fairly soon.

December 3, 2015 at 1:30 pm

13.3.C. EXERCISE. is missing a period, and instead has an end of proof box.

December 17, 2015 at 12:28 pm

Thanks David! Now fixed. The end of proof box was deliberate (that ends the proof of the theorem), so I’ve now added a sentence to make clear that we are done the proof.

December 3, 2015 at 5:18 pm

It seems to me that after 26.1.7.1 there are a bunch of that should be without the .

December 17, 2015 at 12:19 pm

Quick reality check: all four times?

January 3, 2016 at 8:48 am

Sorry for the late response — I lost the paper on which I had worked this out. I should have been more specific: I think that all the beginning in the unnumbered display right after 26.1.7.1 and to the end of the proof — so, four occurrences — should be .

January 3, 2016 at 8:48 am

Thanks Hoot! I will fix this now. (February 2, 2016: finally fixed!)

December 3, 2015 at 5:48 pm

Hey Ravi,

As always, I am very thankful that you take the time to read these comments/suggestions/corrections!

**Comments:**

1) I think one of the most beautiful interactions between geometry and arithmetic is the fact that Brauer-Severi varieties are classified by . This is hard to discuss in generality without a serious diversion in your text. That said, the most baby case of this can be done already using the material covered in the section on `genus 0 curves’ which is not too long.

Namely, you’ve basically shown that the map from isomorphism classes of genus 0 curves to the Brauer group, sending the conic to the Quaternion algebra is an injection.

2) In 19.5.5 you discuss how discussion of hyperelliptic curves shows that there is a curve of every genus. This is certainly nice, but gives the impression that one needs such complicated considerations to obtain curves of all genus. It might be nice to mention that one can easily write down a curve of every genus in .

3) I think it might be nice to yuck up the preamble to Theorem on Formal Functions more. Namely, proper base change holds under pretty simple hypotheses—proper and Hausdorff?—because in such cases we can zoom in far enough to obtain the fiber.

Less cryptically, we know that is the sheafification of . In particular, this presheaf has the same stalk as . In particular, one might imagine that as being the cohomology of as goes over `shrinking neighborhoods’ of the fiber . Of course, the topology on is too coarse to be able to shrink close enough to actually have this limit (over smaller and smaller ) become . But, what formal functions says is that we can *can* zoom in far enough if we’re willing to look formally locally.

**Requests:**

1) There is a topic that is not talked about in modern texts (that I am aware of!), and is constantly used. It would be great to have someone (maybe you 🙂 ) write it up (at least as a guided exercise). Namely, the genus degree formula for singular curves. This would, perhaps, mention the delta invariant (see e.g. here https://en.wikipedia.org/wiki/Delta_invariant).

Even explaining how one can use the usual exact sequence associated to the normalization to compute the genus of in terms of degree of (and its multiplicity of singularities) would be nice—it’s also fairly intuitive.

2) Even though it’s exceedingly simple, I don’t think the projective version of Noether normalization is written down anywhere (at least that I know of!). Having it in writing might be nice.

3) There is no basic texts that give a rigorous grounding in forms/twists (whichever word your prefer). It would be great to have some text that does this for reference. Of course, I assume you don’t want to prove that fpqc descent is effective for quasiprojective schemes, but at least saying that we can classify (separable) twists/forms using cohomology would be nice.

Even emphasizing how nuanced the functor is (needn’t be injective, nor surjective).

Thanks again!

December 14, 2015 at 10:53 am

I think there’s probably a little bit more to say in the proof of 18.1.9. To the beginner I don’t think it’s totally clear that is affine over . There’s this whole argument about looking at a distinguished affine of contained in and then noting that the preimage of that is distinguished in the affine .

It might also be good to just reduce to the case of right at the start to make the notation less weird.

December 14, 2015 at 10:53 am

Sorry, I guess \operatorname doesn’t work here. Hopefully the comment is still understandable.

December 17, 2015 at 12:19 pm

Thanks Hoot! I’ve patched the latex, and will take a look at this (and earlier comments you’ve made that are very useful).

December 22, 2015 at 2:39 pm

It should be clear what is meant, but strictly speaking Exercise 21.2.U probably needs more notation — is already a sheaf on so pulling it back under seems wrong.

December 29, 2015 at 4:57 pm

Thanks! I’d caught this, but had only labeled it with a note “to be fixed” to myself. (It is the 278th such “todo” note in the current version!)

December 25, 2015 at 8:59 am

In 5.5.M(c), when you say “minimal prime” do you mean “associated prime”? [Happy holidays!]

December 25, 2015 at 5:34 pm

Thanks! Now fixed. And happy holidays to you too!

December 31, 2015 at 1:19 pm

In 9.1.C, when you say

…of the “” of all functors…

I think you meant

…of the “functor category” of all functors…

(Interestingly, this was fixed in an older version but not this one). This book is fantastic, by the way—this is just a small typo I noticed.

February 2, 2016 at 9:53 am

Thank you! It is now fixed.

December 31, 2015 at 4:06 pm

In 9.1.D, when you say

Do not use the fact that is representable!

I think you meant

Do not use the fact that is representable!

February 2, 2016 at 9:54 am

Thanks for this too — now fixed!