Let $J$ be the hyperplane divisor in $\mathbb{C}P^2$, and let $i:C \hookrightarrow \mathbb{C}P^2$ be the closed immersion of a smooth generic curve of degree 2. We know that $C\simeq \mathbb{C}P^1$, and let $H$ be the hyperplane divisor in $C$. So loosely $H \sim \frac{1}{2}J|_C$. The question is, what is the projective (locally free) resolution of $i_* \mathcal{O}_C(H)$? in other words, we have the following resolution in general,

\begin{equation} 0\longrightarrow \mathcal{L}_2\longrightarrow\mathcal{L_1}\longrightarrow\mathcal{L}_0\longrightarrow i_* \mathcal{O}_C(H) \longrightarrow 0, \end{equation}

what are the locally free sheaves $\mathcal{L}_i$?