Content deleted Content added
major reorganization and expansion |
→Blowing up submanifolds in complex manifolds: added paragraph on transforms |
||
Line 26:
Since <math>E</math> is a smooth divisor, its normal bundle is a [[line bundle]]. It is not difficult to show that <math>E</math> intersects itself negatively. This means that its normal bundle possesses no holomorphic sections; <math>E</math> is the only smooth complex representative of its [[homology (mathematics) | homology]] class in <math>\tilde X</math>. (Suppose <math>E</math> could be perturbed off itself to another complex submanifold in the same class. Then the two submanifolds would intersect positively — as complex submanifolds always do — contradicting the negative self-intersection of <math>E</math>.) This is why the divisor is called exceptional.
Let <math>V</math> be some submanifold of <math>X</math> other than <math>Z</math>. If <math>V</math> is disjoint from <math>Z</math>, then it is essentially unaffected by blowing up along <math>Z</math>. However, if it intersects <math>Z</math>, then there are two distinct analogues of <math>Z</math> in the blow-up <math>\tilde X</math>. One is the '''proper''' (or '''strict''') '''transform''', which is the closure of <math>\pi^{-1}(Z \setminus E)</math>; its normal bundle in <math>\tilde X</math> is typically different from that of <math>Z</math> in <math>X</math>. The other is the '''total transform''', which incorporates some or all of <math>E</math>; it is essentially the pullback of <math>Z</math> in [[cohomology]].
==Blowing up schemes==
| |||