6
$\begingroup$

Let $X$ be a hypersurface in $\mathbb P^{n}$ defined by the vanishing set of a homogeneous degree $k$ polynomial.

Why is the sequence

$0 \rightarrow \mathcal O(-k) \rightarrow \mathcal O_{\mathbb P^{n}} \rightarrow i_{*} \mathcal O_{X} \rightarrow 0$

(where $i_{*}$ is the inclusion of $X$ into $\mathbb P^{n}$)

exact?

I have seen this or a very similar statement (slightly more/less general) referenced in several sources including Hartshorne and it is always stated as fact.

2 Answers 2