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Let $V\subset\mathbb{P}^n$ be a projective variety and $C_V$ its conormal subvariety in $T^\ast\mathbb{P}^n$. Denote by $\mathscr{O}_{C_V}$ its structure sheaf, then when will the condition

$\mathit{Ext}^1(\mathscr{O}_{C_V},\mathscr{O}_{C_V})=0$

hold for $V$?

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    $\begingroup$ Are you asking about $\text{Ext}^1_{\mathcal{O}_{C_V}}$, or are you asking about $\text{Ext}^1_{\mathcal{O}_{T*\mathbb{P}^n}}$? For the former, quite frequently the Ext group will be zero, e.g., when $V$ is any rational curve. However, for the latter, the group is nonzero already when $V$ is a line in $\mathbb{P}^2$. $\endgroup$
    – Jason Starr
    Aug 27, 2015 at 12:38

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