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Let $C$ be a smooth curve on a smooth quadric $Q$ in $\mathbb P^3$. I have read in Hartshorne's Deformation Theory that the normal bundle of $C$ in $Q$ is $\mathcal N_{C/Q}\cong\mathcal O_C(C^2)$. Could you give me some hint on why is this true? Actually, I thought that the normal bundle should be $\mathcal O_C(C)$.

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    $\begingroup$ This looks just to be a notational issue. $\mathcal O_C(C)$ is a little imprecise if we're talking sheaves on $C$; that one is really thinking of it as a sheaf on $Q$ (supported on $C$). $C^2$ is the self-intersection divisor of $C$, the set of points on $C$ that you twist by to get the normal sheaf. $\endgroup$ – John Brevik Feb 20 '18 at 16:31
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As John Brevik explained, $C^2$ here means a divisor, and not a number.

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