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.



Which projective varieties cannot appear as an exceptional divisor of a blow-up of the projective space
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.


