I just did a "major reorganization and expansion", so I apologize for stepping on toes. I'm being bold.
I'm most comfortable with blow-ups in the smooth complex category, so somebody with broader knowledge could probably connect the scheme-theoretic part better to the rest of the article. In particular, I'm not sure what conditions on and are needed for deformation to the normal cone.
I claim that the exceptional divisor is "exceptional" because it's isolated; but perhaps it's exceptional because it's not in the domain of the birational map? I don't really know.
I think I've heard that any birational map factors as a sequence of blow-ups and blow-downs. That's worth mentioning (perhaps in Birational geometry, where "birational map" should really be defined).
I started all of this because someone requested an article on "Exceptional divisor". However, it is only referenced in the Cubic surface article, so I think I'll connect that link to here instead. Joshuardavis
Just wondering, Blowing up can also refer to the act of exploding something - does anyone else agree that this is a disambiguation or at least worthy of a link? --Pcmattman (talk) 10:14, 15 May 2008 (UTC)
- You're right; there should be a disambiguation link to Explosion, say. But I think that there should not be a whole disambiguation page. Joshua R. Davis (talk) 12:42, 15 May 2008 (UTC)