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.
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2$\begingroup$ Is not the exceptional divisor always covered by rational curves (because it is the Proj of the normal sheaf)? This should exclude all the non-uniruled varieties, I think. $\endgroup$– Francesco PolizziJun 26, 2021 at 18:45
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$\begingroup$ @FrancescoPolizzi By your argument that it is the Proj of a normal sheaf, can't we further say that the irreducible components of the exceptional divisor must be rational? $\endgroup$– RonJun 27, 2021 at 9:20
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$\begingroup$ unirational, but not rational in general. If you blow-up a smooth curve $C$ of genus $g\geq 1$ in $\mathbb{P}^3$, the exceptional divisor is a $\mathbb{P}^1$-bundle over $C$, that is not rational because it has irregularity $g$. $\endgroup$– Francesco PolizziJun 27, 2021 at 9:25
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$\begingroup$ @FrancescoPolizzi Thanks. You could write your comments as the answer. This does answer my question to a large extent. $\endgroup$– RonJun 27, 2021 at 9:28
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$\begingroup$ Ok, I wrote a short answer $\endgroup$– Francesco PolizziJun 27, 2021 at 9:46
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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.
answered Jun 27, 2021 at 9:45