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:
“The wind of change is blowing through the continent. Whether we like it or not, this growth of national consciousness is a political fact.”
—Harold MacMillan (18941986)
“A bath and a tenderloin steak. Those are the high points of a mans life.”
—Curtis Siodmak (19021988)
“The money complex is the demonic, and the demonic is Gods ape; the money complex is therefore the heir to and substitute for the religious complex, an attempt to find God in things.”
—Norman O. Brown (b. 1913)
“It is the space inside that gives the drum its sound.”
—Hawaiian saying no. 1189, lelo NoEau, collected, translated, and annotated by Mary Kawena Pukui, Bishop Museum Press, Hawaii (1983)