If $Y$ is a complete intersection in $\mathbb{P}^{n}$ of codimension $r$, $Y=H_{1} \cap \dots \cap H_{r}$, where $H_{i}$'s are hypersurfaces of degree $d$. Let $Z=H_{1} \cap \dots \cap H_{r-1}$.
Why there exists the short exact sequence: $0 \to \mathcal{O}_{Z}(n-d) \to \mathcal{O}_{Z}(n) \to \mathcal{O}_{Y}(n) \to 0$?
Thanks!
The closed embedding $Y \hookrightarrow Z$ induces a surjective map on sheaves $\mathcal{O}_Z \rightarrow \mathcal{O}_Y$, whose kernel is precisely the set of regular functions on $Z$ vanishing on $Z\cap H_r$, i.e. $\mathcal{O}_{Z}(-d)$. So we have an exact sequence $$0 \rightarrow \mathcal{O}_Z(-d) \rightarrow \mathcal{O}_Z \rightarrow \mathcal{O}_Y \rightarrow 0$$ We can tensor this with $\mathcal{O}_Z(n)$ and it will stay exact since it locally stays exact because $\mathcal{O}_Z(n)$ is invertible.