map between and Cn. If instead we consider the holomorphic projection we obtain the tautological line bundle of and we can identify the exceptional divisor with its zero section, namely which assigns to each point the zero element in the fiber over . Blowing up submanifolds in complex manifolds More generally, one can blow up any codimension-k complex submanifold Z of Cn. Suppose that Z is the locus of the equations , and let be homogeneous coordinates on Pk - 1. Then the blow-up is the locus of the equations for all i and j, in the space Cn × Pk - 1. More generally still, one can blow up any submanifold of any complex manifold X by applying this construction locally. The effect is, as before, to replace the blow-up locus Z with the exceptional divisor E. In other words, the blow-up map is a birational mapping which, away from E, induces an isomorphism, and, on E, a locally trivial fibration with fiber Pk - 1. Indeed, the restriction is naturally seen as the projectivization of the normal bundle of Z in X. Since E is a smooth divisor, its normal bundle is a line bundle. It is not difficult to show that E intersects itself negatively. This means that its normal bundle possesses no holomorphic sections; E is the only smooth complex representative of its homology class in . (Suppose E 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 E.) This is why the divisor is called exceptional. Let V be some submanifold of X other than Z. If V is disjoint from Z, then it is essentially unaffected by blowing up along Z. However, if it intersects Z, then there are two distinct analogues of V in the blow-up . One is the proper (or strict) transform, which is the closure of ; its normal bundle in is typically different from that of V in X. The other is the total transform, which incorporates some or all of E; it is essentially the pullback of V in cohomology. Blowing up schemes To pursue blow-up in its greatest generality, let X be a scheme, and let be a coherent sheaf of ideals on X. The blow-up of X with respect to is a scheme