# Resolution of a torsion sheaf

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,

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

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

• It is possible to put some topological constraints on $\mathcal{L}_i$'s, but I'm curious to see whether it is possible to say more.... Commented Dec 3, 2018 at 5:24

## 1 Answer

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.