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.

blowup

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.)

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