This is a bonus post. I’ve edited the chapter on blowing up, and as this is a starred optional chapter (not required reading), I’m posting it now. It is Chapter 19 of the March 16 version posted at the usual place.

If you always hated blowing up, I hope this might make you reconsider.

Here is what I do. Of course, I start (in 19.1) with the “baby example” of blowing up a point in the plane. Then (19.2) I find the universal property of blowing up to be surprisingly useful, and not at all frightening, although it doesn’t clearly have any relationship to the special case blowing up the origin in the plane. The “blow-up closure lemma” (19.2.6) is one I wish I knew when I was younger. (For example, when resolving plane curve singularities by blowing up the plane and taking the proper transform, I didn’t realize that the proper transform was automatically a blow-up of the curve!) Then in 19.3 I finally give the Proj description of the blow-up, only after the special case and the universal property. (Yes, I realize I could have done this in line 1 of the chapter.) Finally, as always, the most important part of learning a concept is actually using it, so in 19.4, I have a large number of specific computations to hopefully convince you that you can really work with this concept.

*For learners*

You should read this section only if you really want to, but if you do, I’d very much like to hear your thoughts. What was hard? What worked and what didn’t work in the exposition?

*For experts*

I mention many examples that people might see in one way or another. On Mathoverflow, someone asked if I could include the deformation to the normal cone, so I did. The fun small resolution is there. But far more “basic” examples are there too. If there is any basic example missing, please let me know!

I had an interesting discussion with Brian Conrad on blow-ups (witness his comments on the previous post). He has pointed out that I can give another definition that is sometimes used. I’ve chosen not to, because I fear it will just confuse an already confusing situation. What I’ve described is enough to do all the exercises, and the description he mentioned to me likely will be relevant for only a small part of the community, and that part of the community will be able to learn it basically immediately on their own, given what I’ve written.

*For everyone*

As with many other ideas in algebraic geometry, blowing up led to a movie — see the poster above. I haven’t actually seen the movie, but from what I’ve read online, it hews closely to the mathematics. “Blowing up” means zooming in. While easy to say in general terms, it involves some work and technique. And if you really want to understand a situation, you have to blow up repeatedly, until the situation is resolved.

(Corrections and suggestions from a huge number of people continue to be implemented. Many are from the comments on this site, but many aren’t, so I want to mention some names here, but I’m forgetting a bunch I’m sure: Chris Davis, Jason Ferguson, … . I remain behind on responses, so I’m trying to be systematic in my responses, dealing with various things on a single topic all at once.)

March 24, 2011 at 1:13 pm

I know this goes back a few chapters, but we’ve been rather confused about exercise 15.2R lately. Is it a typo? It seems like in that ring, we have x^2 = (z-y)(z+y), and I don’t see how these factorizations are equivalent. (We also think there are problems with the Zariski tangent space at the origin being of the wrong dimension.) For the same reason, we’re also confused by the statement 15.2S seems to make about k[x,y,z]/(x^2 + y^2 + z^2) being a UFD even if k is algebraically closed.

We were trying to solve 15.2R with (x^2 + y^2 – z^2) replaced by (x^2 + y^2 + z^2), but couldn’t make much progress on the latter.

March 25, 2011 at 6:19 am

Hi Silas,

(Most people are at various random points in the notes, so this isn’t really a few chapters ago…)

You’re right — this is more than a typo, it is a mangling of what I’d intended to do. I will fix this, hopefully soon (in time for the next posting on Apr. ~2). Here is the intended story. I’ve not thought through the exposition, but here is the content.

First, just after 15.2.P, there should be the following exercise. k is a field of char not 2. We’ll figure out the class group of k[x,y,z]/(x^2+y^2+z^2).

(a) If k does not contain a square root of -1, then consider Z=V(z). It is irreducible. Use (15.2.6.2) (an “excision exact sequence” for class groups). Note that [Z] = 0 in the class group, and Cl(X-Z)=0 (I think X-Z is \A^2 or an open subset thereof). So Cl(X)=0.

(b) If k contains a square root of -1, then basically do the same thing. Z now has 2 components, say Z_1 and Z_2, so (15.2.6.2) turns into

\Z \oplus \Z –> Cl X –> Cl(X-Z) –> 0. The same argument as in (a) shows that [Z_1] + [Z_2]=0, and a similar argument shows 2 [Z_1] = 2 [Z_2]=0. Also Cl(X-Z)=0 as in (a).

But an earlier exercise, that the ruling on the cone is

notprincipal, implies that Cl X is not trivial. Thus Cl X = Z/2.Then 15.2.R will instead be the following: if k is a field of characteristic not 2 then k[x,y,z]/(x^2+y^2+z^2) is a UFD if and only if k does *not* contain a square root of -1. (Your argument that k[x,y,z]/(x^2+y^2-z^2) is not a UFD can be made completely precise — but to follow up on your question, \R[x,y,z]/(x^2+y^2+z^2) *is* a UFD!)

Then the discussion after 15.2.S changes as follows: Consider k[x1, …., xn] / (x1^2 + … + xm^2) where k is algebraically closed. The cases m=1 and m=2 are special cases. (If m=1, we get something nonreduced. If m=2, we get something reduced but reducible.) If m=3 and 4 we get domains that do not have unique factorization. And then 15.2.T shows that once m is at least 5, we always get UFD’s. We now have an interesting question (that someone may be able to answer). When m=3, whether the ring is a UFD depends on precisely whether -1 has a square root. For m at least 4, *if* -1 has a square root, then the above statements hold. What happens if -1 does not have a square root? I think that we’re still fine in m >= 5 (think this through!), and that to answer the question for m=4, we need to think about the Galois action on the Picard group of the quadric surface.

I want to think this through later. (It is a nice mix of algebra, geometry, and arithmetic.)

March 31, 2011 at 8:08 am

A follow-up: part of what I said in my previous reply was garbage. In the new notes (to be posted later today), I’ll just discuss the case where k is algebraically closed and characteristic not 2. Part (b) in the previous reply was garbage, because it is *not* true in general that the complement of a ruling (if -1 is not a perfect square) is an open subset of the plane. Here’s an example where it is not true: let k be the real numbers (one of the few times when I’m not frightened of the real numbers). Then x^2+y^2+ z^2 = 0 minus a ruling has no k-points — there are no values of x,y,z, not all zero, such that x^2+y^2+z^2=0!

I later have a remark discussing this interesting case (Martin Olsson pointed out how to deal with it over dinner on Tuesday), but it is beyond the scope of these notes.

March 31, 2011 at 7:28 am

[…] I’ve not looked over Chapter 19 in a very long time, so look at it at your own risk. (But comments are still welcome of course!) Update: it has now been edited, see this March 16 post. […]

June 15, 2011 at 8:23 pm

In doing exercise 19.2.G, I realised the following (hopefully this is correct): In section 19.2.5, the top square of the last diagram is actually a blow-up square, by exercises 19.2.E and F and the fact that is a locally principal subscheme of , as well as the fact that blow-ups and scheme-theoretic closures (in this case) can be computed affine-locally. It would be helpful if this were pointed out in the notes.

The diagram in ex 19.2.G would be more helpful if the rectangle which has two sides marked as closed immersions were split into two squares, as in the last diagram of section 19.2.5 I referred to above. Are the morphisms marked as closed immersions in 19.2.G necessarily closed immersions (if we aren’t assuming is a closed immersion)?

March 27, 2012 at 7:56 pm

Long-delayed response (sorry!!!): thanks for this! Regarding the first paragraph: I’ve pointed this out in the notes. And regarding the second paragraph: since then (and before now), I’ve changed the diagram so it is more of a parallelopiped, in a way which likely addresses the issue you pointed out.