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)
“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)
“Only that which points the human spirit beyond its own limitations into what is universally human gives the individual strength superior to his own. Only in suprahuman demands which can hardly be fulfilled do human beings and peoples feel their true and sacred measure.”
—Stefan Zweig (18811942)
“The human mind is so complex and things are so tangled up with each other that, to explain a blade of straw, one would have to take to pieces an entire universe.... A definition is a sack of flour compressed into a thimble.”
—Rémy De Gourmont (1858–1915)
“True spoiling is nothing to do with what a child owns or with amount of attention he gets. he can have the major part of your income, living space and attention and not be spoiled, or he can have very little and be spoiled. It is not what he gets that is at issue. It is how and why he gets it. Spoiling is to do with the family balance of power.”
—Penelope Leach (20th century)