$\mathcal{O}_{X}(d)\simeq \mathcal{O}_{X}(D)$?

Question

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?

$\mathcal{O}_{X}(d)\simeq \mathcal{O}_{X}(D)$ , $P \mapsto \frac{P}{F}$

Answer 1

The isomorphism you've written is certainly generally true, but the global sections of $\mathcal{O}_X(d)$ are not necessarily just the homogeneous polynomials. As a general comment, sheaves are generally not determined by their global sections; many sheaves have none!

As a simple counter-example, let $X$ be three points in $\mathbb{P}^1$, i.e., $X$ is given by the vanishing of a general cubic. Let $D$ be a single point, i.e., a degree-one 'hypersurface'. Then $\mathcal{O}_{\mathbb{P}^1}(D) \cong \mathcal{O}_{\mathbb{P}^1}(1)$, and the space of global sections is two-dimensional. But if we restrict this line bundle to $X$, obviously the space of global sections is three-dimensional (we can choose the value independently over each of the three points of $X$).

In general, if $\imath :X \hookrightarrow \mathbb{P}^n$ is the embedding of $X$, then there is a short exact sequence: $$ 0 \longrightarrow \mathcal{I}_X \longrightarrow \mathcal{O}_{\mathbb{P}^n} \longrightarrow \imath_*\mathcal{O}_X \longrightarrow 0 ~, $$ where $\mathcal{I}$ is the ideal sheaf of $X$ (the sheaf of sections of $\mathcal{O}_{\mathbb{P}^n}$ which vanish on $X$). We can twist this by $\mathcal{O}_{\mathbb{P}^n}(d)$ to get $$ 0 \longrightarrow \mathcal{I}_X(d) \longrightarrow \mathcal{O}_{\mathbb{P}^n}(d) \longrightarrow \imath_*\mathcal{O}_X(d) \longrightarrow 0 ~, $$ Since $d>0$, we have $H^1(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(d)) =0$, so the long exact sequence in cohomology tells us that the space of global sections of $X$ is given by the ones you wrote down, plus some others, which are counted by $H^1(\mathbb{P}^n,\mathcal{I}_X(d))$. The calculation of this group depends a lot on the case you consider.

In my simple example, $\mathcal{I}_X(d) \cong\mathbb{P}^1(-2)$, and $H^1(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(-2))\cong \mathbb{C}$, giving the extra section.