Exceptional divisor ($E$)

The exceptional divisor of the blowup of a complex manifold at a point $p$ is the fiber over $p$. In complex dimension $n$ it is naturally the projective space $\mathbb{P}(T_{p}X)\cong {\mathbb{CP}}^{n-1}$ of tangent directions at the center.

For the blowup of a complex manifold at a point in dimension $n\ge 2$, the exceptional divisor is ${\mathbb{CP}}^{n-1}$ and $[E]{|}_E={\mathcal{O}}_E(-1)$. The local blowup is the total space of the tautological line bundle, so its zero section has that normal bundle. Restricting a power to a projective line inside $E$ gives degree $-r$; a trivial bundle has degree zero. This proves that no nonzero power of $[E]$ is trivial, without assuming global projectivity of the original manifold.