On $\mathbb{P}^n$ let $D$ be a smooth hypersurface defined by the equation $F=0$, F an homogeneous polynomial.
$\mathcal{O}_{\mathbb{P}^n}(D)$ is the sheaf of meromorphic functions on $\mathbb{P}^n$ with poles on $D$.
$\mathcal{O}_{\mathbb{P}^n}(d)$ , $d > 0$ is the sheaf of the sections of the fiber bundle ${S^*}^{\bigotimes d}$, $S$ is the tautological bundle of $\mathbb{P}^n$.these sections are given by the homogeneous polynomials of degree d. We have the isomorphism
$\mathcal{O}_{\mathbb{P}^n}(d)\simeq \mathcal{O}_{\mathbb{P}^n}(D)$ , $P \mapsto \frac{P}{F}$
I wonder if the same thing values for a generic projective variety $X$ and $D\subset X$ a smooth hypersurface defined by $F=0$.$\mathcal{O}_{X}(d)$ , $d > 0$ will be the sheaf of the sections of the fiber bundle ${S^*}^{\bigotimes d}$, $S$ is the tautological bundle on X. These sections are given by the homogeneous polynomials of degree d in $\mathbb{C}[X_0, \cdots , X_n]/I$ , $I$ defining ideal of $X$, am i right?
Does this isomorphism still stand?
$\mathcal{O}_{X}(d)\simeq \mathcal{O}_{X}(D)$ , $P \mapsto \frac{P}{F}$