This should be an elementary question: say if I have a plane conic $C\subset \mathbb{P}^2$, then the ideal sheaf is $\mathcal{I} = \mathcal{O}_{\mathbb{P}^2}(-2)$. How do I calculate $\mathcal{N}=\mathcal{I}/\mathcal{I}^2$? Is there a general way to calculate such $\mathcal{N}$?

  • $\begingroup$ What do you mean, "calculate"? What form of answer are you looking for? $\endgroup$
    – KReiser
    Jun 26, 2020 at 18:23
  • $\begingroup$ @KReiser Say it can be expressed as $\mathcal{O}_{\mathbb{P}^2}(m)$ and $\mathcal{O}_{C}(n)$, I want to know what $m,n$ are. $\endgroup$ Jun 26, 2020 at 18:28
  • $\begingroup$ In this case $N$ is just restriction of $I$ to $C$ and thus $n=-2$. Typically one uses $N$ for the dual in your situation. $\endgroup$
    – Mohan
    Jun 26, 2020 at 19:16
  • $\begingroup$ @Mohan Could you explain in more detail? I don't understand why this is the restriction of $I$ to $C$... $\endgroup$ Jun 26, 2020 at 20:32
  • 1
    $\begingroup$ You have an exact sequence, $0\to I^2\to I\to I/I^2\to 0$. Notice that $I=O(-2)$ and then $I/I^2=O(-2)_{|C}=O_C(-2)$. $\endgroup$
    – Mohan
    Jun 27, 2020 at 21:42


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