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I learnt that the dimension of the blowup fibers “explodes” in the center of the blowup and my imagination for now goes as follows: every line on the affine plane is sent identically to a line in the blowup, except for all the lines passing through the center at which they seem to “explode”. Still, I’d like to have some more geometric intuition, if available, about what is happening here: why does this transformation provide any help for algebraic geometry? Why is it so fundamental to the nature of birational maps (to the point that, for example, every birational map can be factorized into simple blowups)? Quoting from Wikipedia page “the metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion“, which confused me even more. Any help would be very appreciated.

G. Chiusole
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cip
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    Intuition for blowups has been asked about before on MSE here and here. Some of this may be helpful to you. – KReiser Sep 09 '20 at 20:24
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    I would like to note that blow ups aren't a reference to blowing things up like an explosion but rather blowing things up like a balloon. It might help your intuition. – Asvin Sep 09 '20 at 22:36
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    @Asvin It's interesting to note that in French, blow-ups (in math) are called "éclatements" from the verb "éclater" which can be translated by burst or break. I wonder who introduced blow-ups in a first place. – Dominique Mattei Sep 10 '20 at 07:42
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    Same thing in Italian, blow-ups are known as “scoppiamenti”, which also translates as “burst” – cip Sep 10 '20 at 08:19
  • Very interesting! Maybe the original intuition was indeed modeled after an explosion! – Asvin Sep 10 '20 at 12:44

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