- Theorem :
For $ X $ a smooth variety ( or complex manifold ), and $ Y \subseteq X $ of codimension $ 1 $, then : $$ K_Y = (K_X \otimes \mathcal{O}_X (Y))_{|Y} $$
- Questions :
$ 1) $ What is the meaning of $ \mathcal{O}_X (Y) $ in the formula : $ K_Y = (K_X \otimes \mathcal{O}_X (Y))_{|Y} $ ?
$ 2) $ Suppose $ X \subseteq \mathbb{P}^2 $ is a smooth curve of degree $ d $. Why is : $ \mathcal{O}_{ \mathbb{P}^{2} } (X) = \mathcal{O}_{\mathbb{P}^{n}} (d) $ ?
Thanks in advance for your help.