Resolution of a torsion sheaf

Question

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 $\mathcal{L}_i$?

Answer 1

Write the equation of $C$ as $\ XY-Z^2=0$, and consider the homomorphism $u: \mathcal{O}_{\mathbb{P}^2}(-1)^2\rightarrow \mathcal{O}_{\mathbb{P}^2}^2$ given by the matrix $\begin{pmatrix} X & Z\\Z& Y \end{pmatrix}$. It is invertible outside $C$, and has rank 1 at every point of $C$. Thus $u$ is injective, and its cokernel is $i_*L$, where $L$ is a line bundle on $C$. The exact sequence $0\rightarrow \mathcal{O}_{\mathbb{P}^2}(-1)^2\xrightarrow{\ u\ } \mathcal{O}_{\mathbb{P}^2}^2\rightarrow i_*L\rightarrow 0\ $ gives $h^0(L)= 2$, thus $L=\mathcal{O}_C(H)$, and this exact sequence gives the resolution you are asking for.