# Calculating the sheaf $\mathcal I/\mathcal I^2$

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 $$\mathcal{N}$$?

• What do you mean, "calculate"? What form of answer are you looking for? Jun 26, 2020 at 18:23
• @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. Jun 26, 2020 at 18:28
• 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. Jun 26, 2020 at 19:16
• @Mohan Could you explain in more detail? I don't understand why this is the restriction of $I$ to $C$... Jun 26, 2020 at 20:32
• You have an exact sequence, $0\to I^2\to I\to I/I^2\to 0$. Notice that $I=O(-2)$ and then $I/I^2=O(-2)_{|C}=O_C(-2)$. Jun 27, 2020 at 21:42