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 up, blowing, points, complex and/or space:
“The wind speeds her,
Blowing upon her hands
And watery back.
She touches the clouds, where she goes
In the circle of her traverse of the sea.”
—Wallace Stevens (1879–1955)
“Sleep, ignorant of pain, sleep, ignorant of grief, may you come to us blowing softly, kindly, kindly come king.”
—Sophocles (497–406/5 B.C.)
“Every man has to learn the points of the compass again as often as he awakes, whether from sleep or any abstraction.”
—Henry David Thoreau (1817–1862)
“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)
“For good teaching rests neither in accumulating a shelfful of knowledge nor in developing a repertoire of skills. In the end, good teaching lies in a willingness to attend and care for what happens in our students, ourselves, and the space between us. Good teaching is a certain kind of stance, I think. It is a stance of receptivity, of attunement, of listening.”
—Laurent A. Daloz (20th century)