I am looking for examples of projective varieties (over $\mathbb{C}$) of dimension, say $n$ which cannot appear as an exceptional divisor of a blow-up of $\mathbb{P}^{n+1}$ along some closed subscheme. Any idea/reference will be most welcome.

## 1 Answer

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The exceptional divisor is the $\operatorname{Proj}$ of the normal sheaf, hence it is covered by rational curves. This excludes all the non-uniruled varieties.

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