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.
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