= Nontrivial powers of the exceptional line bundle of a point blowup
{title2=$[E]^{\otimes r}\text{ is trivial}\Longleftrightarrow r=0$}
For the <blowup of a complex manifold at a point> in dimension $n\ge2$, 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.
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