complex numbers, the blowup has a topological description as the connected sum . Assume that P is the origin in A2 ⊆ P2, and write L for the line at infinity. A2 \ {0} has an inversion map t which sends (x, y) to (x/(|x|2 + |y|2), y/(|x|2 + |y|2)). t is the circle inversion with respect to the unit sphere S: It fixes S, preserves each line through the origin, and exchanges the inside of the sphere with the outside. t extends to a continuous map P2 \ {0} → A2 by sending the line at infinity to the origin. This extension, which we also denote t, can be used to construct the blowup. Let C denote the complement of the unit ball. The blowup X is the manifold obtained by attaching two copies of C along S. X comes with a map π to P2 which is the identity on the first copy of C and t on the second copy of C. This map is an isomorphism away from P, and the fiber over P is the line at infinity in the second copy of C. Each point in this line corresponds to a unique line through the origin, so the fiber over π corresponds to the possible normal directions through the origin. For CP2 this process ought to produce an oriented manifold. In order to make this happen, the two copies of C should be given opposite orientations. In symbols, X is , where is CP2 with the opposite of the standard orientation. 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