Let $Y\subset X$ be a smooth hypersurface defined by a section $s\in H^0(X,L)$ for some holormorphic line bundle $L\in Pic(X)$.Show that the normal bundle $\mathcal N_{Y/X}$ is isomorphic to $L|_Y$.
I can prove this claim by using the cocycle description,however,how to prove it by using the fact "$\mathcal N_{Y/X}^*\cong\mathcal I_Y/\mathcal I_Y^2$".
Can anyone give me some advice?Thanks a lot!