Blowing Up Points in Complex Space
Let Z be the origin in n-dimensional complex space, Cn. That is, Z is the point where the n coordinate functions simultaneously vanish. Let Pn - 1 be (n - 1)-dimensional complex projective space with homogeneous coordinates . Let be the subset of Cn × Pn - 1 that satisfies simultaneously the equations for i, j = 1, ..., n. The projection
naturally induces a holomorphic map
This map π (or, often, the space ) is called the blow-up (variously spelled blow up or blowup) of Cn.
The exceptional divisor E is defined as the inverse image of the blow-up locus Z under π. It is easy to see that
is a copy of projective space. It is an effective divisor. Away from E, π is an isomorphism between and Cn \ Z; it is a birational map between and Cn.
Read more about this topic: Blowing Up
Famous quotes containing the words blowing, points, complex and/or space:
“But the word Miracle, as pronounced by Christian churches, gives a false impression; it is Monster. It is not one with the blowing clover and falling rain.”
—Ralph Waldo Emerson (1803–1882)
“PLAIN SUPERFICIALITY is the character of a speech, in which any two points being taken, the speaker is found to lie wholly with regard to those two points.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“All propaganda or popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that.”
—Terry Eagleton (b. 1943)
“Oh, my. I’d forgotten how much I hate space travel.”
—George Lucas (b. 1944)