I realize that there needs to be a place for people to make general comments and requests, so here it is. If there seems to be a call for more specific groupings of comments, I’ll add pages for such things.

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September 3, 2010

I realize that there needs to be a place for people to make general comments and requests, so here it is. If there seems to be a call for more specific groupings of comments, I’ll add pages for such things.

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September 3, 2010 at 10:04 pm

I’ll kick this off with an email from Kamal Khuri-Makdisi (copied with his permission):

Hello Ravi,

I came across your worldwide reading course on algebraic geometry. It’s an amazing idea! I thought I would share with you two elementary pedagogical suggestions, (0) and (1):

0) if you get around to discussing blowups in the notes: in my youth, I remember being mystified by the “gluing” construction of a blowup — what on earth was a P^{n-1} doing being glued into, say, A^n – 0? Only much later did I realize that it was really taking some (quasi-)projective embedding (say of A^n) via a linear series (say of polynomials of a fixed large degree), and then looking at the linear subseries of sections that vanish at 0. Due to the base point, the subseries cannot give a morphism from A^n to projective space, but the blowup is the closure of the image of A^n – 0 and the incomplete linear series becomes “nice” on the blowup, related to a divisor as opposed to a smaller subvariety. I wish this had been mentioned clearly in Hartshorne or Griffiths-Harris back then.

1) on the structure sheaf of Spec A, or more generally for the quasi-coherent sheaf coming from an A-module M:

The usual proof that the presheaf is in fact a sheaf uses a “partition of unity” argument, which you give in an earlier version of your notes. There is an alternative method that you might call an “annihilator” argument, that is worth giving as a second proof or including as an exercise.

More precisely, say you have a cover by Spec A_{f_i} and have a compatible collection of x_i in A_{f_i} that you want to glue to get an element of A. Define an ideal

I = {r in A : there exists y in A such that r x_i = y/1 in every A_{f_i}}.

Now show (by the compatibility of the x_i’s) that I contains some power of each f_i, and so contains 1. The point is that I is the annihilator of the tuple (x_i) in (\prod_i A_{f_i})/diag(A). An even easier “annihilator” argument (basically the standard argument) shows that the diagonal map from A to the product is injective.

I assume this already appears somewhere, but I stumbled across it independently some years ago while thinking about a commutative algebra course I was teaching at the time. I basically imitated the “descent” proof of exactness of the sequence

0 -> A -> \prod_i A_{f_i} -> \prod_{i,j} A_{f_i f_j}

and combined this with the details of the proof that it is enough to check exactness after tensoring with the faithfully flat module B = \prod_i A_{f_i}.

Best regards,

– Kamal Khuri-Makdisi

March 14, 2011 at 11:07 am

Also: if anyone particularly likes (or dislikes) Kamal’s point of view, please express an opinion!

March 17, 2011 at 12:47 am

For 0 I had the same doubt but for me there is no better picture than the one in Hartshorne, chapter I.

Also, I do not understand (and I would like to see it in the notes, although I guess it is far too specific for this course) why blowing up 5 points in general position allows you to embed the resulting space in P^4 but doing so with 6 allows you to embed the resulting space in P^3.

March 17, 2011 at 9:14 am

Here is a baby example which may help make that reasonable. Say I wanted to embed P^2 in P^3 by cubics. You can’t do it: if you take a 4-dimensional linear system of cubics that is base-point-free, the surface is forced to intersect itself. (This can be seen in various ways, none of which are short.) How to remedy this? Well, if you begin to allow yourself a base-locus, you get some benefits; the degree of the image drops, and it turns out that if you have just the right base locus, you can have the surface not intersect itself. But having a base locus kills the chance of it being a map. But you can resolve this map by blowing up the base locus. Thus by blowing up more, you can allow yourself a chance to embed something in P^n.

One upshot is that just because you can blow up P^n in m points and embed it in P^M, you shouldn’t expect this to tell you anything about the analogous problem with same n and M, but different m; the linear series in question can be related, but at the expense of having base loci. (This was written a little hastily, and may not be too comprehensible…)

I should say that at this point in the notes, I haven’t yet even written up an explanation of why P^2 blown up at 6 general points embeds [oops, sorry for the purists: “closedly immerses”] in P^3. I want to at some point.

July 14, 2011 at 8:17 am

This is quite enlightening. It is a shame these comments on the limits of a construction (by limit I mean what happens when the hypothesis are weakened) are hardly ever found in text books

March 17, 2011 at 9:09 am

Kamal, regarding (0), you can take a look at the blow-up notes (now posted). If you have suggestions, of course please let me know!

September 12, 2010 at 8:38 am

Hi,

I was wondering if it would be possible to add a search function to your blog?

As off now, this is not an issue since there are only a few posts, but in sometime it might be hard to recall where to find an specific comment or post.

Great notes by the way.

Thanks

September 12, 2010 at 1:27 pm

If there is some very trivial way for me to do it on wordpress, I’m happy to do it. Otherwise, one easy thing to do is to google “wordpress 216 algebraic geometry [what you’re looking for]”.

September 12, 2010 at 2:33 pm

All you have to do is go to Widgets under appearance on the Dashboard and drag the search form widget to you sidebar on the right.

Thanks.

[Done, thanks! – R.]September 28, 2010 at 7:35 am

Dear Ravi,

Thanks for the notes. We are starting a reading group in Edinburgh based in this material. In the meetings we are intending to discuss some exercises so since we do not know the material in advance, we find really good to have a common set of exercises. Please, keep posting a choice of those you think are interesting. Obviously this does not mean that we do not do others, but at least we make sure that all of us have tried the same.

I cannot guarantee for how long will this reading group live, but I hope it does for a while.

[Great — I’ll keep doing this. If there is something else that might be helpful, please suggest it too. – R.]October 9, 2010 at 3:24 pm

I would like to eventually have better pictures. Mehdi Omidali has made some particularly nice pictures, and kindly emailed them to me, and I will move them in when I have a chance. Until then, I wanted to show them to you all. They are

here (click).October 31, 2010 at 9:42 am

Just saw those pictures… they’re great! Other good candidates for fancy illustrations: [1] Figure 8.7 in the June 15 version of the notes (proof of Chevalley’s constructibility theorem), [2] the proper non-projective surface made using the “banana curve” (rmk. 21.10.8 in June 15 version), and [3] the classical examples of the mystical hexagon and the Poncelet porism, which you mentioned in class but seems not to have made it into the typed notes?

November 5, 2010 at 8:04 pm

Good ideas! (I’ve put flags to myself so I don’t forget.) Once I import the first batch, I may ask Mehdi Omidali if he can make more…

April 1, 2013 at 9:52 am

Mehdi’s pictures are jaw-dropping. The combination of mathematical accuracy and artistic minimalist beauty are unbelievable. I am hoping to see as many pictures of this kind as possible in your book.