Let $X$ be a smooth toric variety and $Y \subset X$ be a complete intersection with the normal bundle $N_{Y/X}$ and $E \subset \mathrm{Bl_Y X}$ be exceptional divisor. Than it is well known that $E \simeq \mathbb{P}(N_{Y/X})$ and $\mathcal{O}_{\mathbb{P}(N_{Y/X})}(-k) \simeq \mathcal{O}_{E}(E)$.
My question is:
How to express $\mathcal{O}_{\mathbb{P}(N_{Y/X})}(-k)$ with $k > 1$ in similar terms?
Actually i want to construct a resolution of $\mathcal{O}_{\mathbb{P}(N_{Y/X})}(-k)$ like $0\to\mathcal{O} \to \mathcal{O}(E)\to \mathcal{O}_{\mathbb{P}(N_{Y/X})}(-1)\to 0$
First, $O_E(E) \cong O_{\mathbb{P}(N)}(-1)$, there is no $k$ yet. From this it is clear that $O_E(kE) \cong O_{\mathbb{P}(N)}(-k)$ for any $k$, and so a possible resolution is $$ 0 \to O((k-1)E) \to O(kE) \to O_{\mathbb{P}(N)}(-k) \to 0. $$