On Extensions of locally free sheaves of rank $2$ on surface

Question

Let $X$ be a projective algebraic surface over $\mathbb C$ and let, $\mathcal F$ be a locally free sheaf of rank $2$,Let's assume $ \mathcal F(1)$ has a section, then is it true that we have an exact sequence of the following form :

$ 0 \to \mathcal O_X \to \mathcal F(1) \to det(\mathcal F) \otimes I_Z \to 0 $

for some zero dimensional subscheme $Z$ in $X$ ?

We can think of a natural map from $\mathcal O_X \to \mathcal F(1)$, but I don't see why should it be injective ? and also why the cokernel looks like this?

Can anybody give me a reference? Or a hint to prove it?

Any help from anyone is welcome

Answer 1

The answer is essentially an application of Proposition 5, page 33, of Friedman's Algebraic Surfaces and Holomorphic Vector Bundles In what follows $V$ is a rank two vector bundle over a complex projective manifold $X$.

Proposition:

1. Let $\phi\colon L \rightarrow V $ be a sub-line bundle (rank one invertible subsheaf). Then there exists an unique effective divisor $D$ on $X$, possibly $0$, such that the map $\phi$ factors through the inclusion $L\rightarrow L \otimes \mathcal{O}_X(D)$ and such that $V/ (L \otimes \mathcal{O}_X(D))$ is torsion free.
2. In the above situation, if $V/L$ is torsion free i.e. $D=0$ then there exists a local complete intersection codimension two scheme $Z$ of $X$ and an exact sequence $$0 \rightarrow L \rightarrow V \rightarrow L'\otimes \mathcal{I}_Z \rightarrow 0 $$

To give a global section $s$ of $\mathcal{F}(1)$ leads to a map $\phi_s \colon \mathcal{O}_X \rightarrow \mathcal{F}(1)$ defined by multiplication $f \mapsto fs$. If it had a nontrivial kernel then $s$ would be a torsion section which is not possible since $\mathcal{F}(1)$ is locally free. Hence $\phi_s$ is injective.

We then have $$ 0 \rightarrow \mathcal{O}_X \rightarrow \mathcal{F}(1) \rightarrow \operatorname{coker}(\phi_s) \rightarrow 0 $$ and if $\operatorname{coker}(\phi_s)$ is torsion free the Proposition tells us that $ \operatorname{coker}(\phi_s) \simeq L \otimes \mathcal{I}_Z$ for some $0$-dimensional subscheme $Z$ (maybe empty) and some linebundle $L$. As $Z$ has codimension two, we may coimpute the determinants from the above sequence to conclude that $$ L \simeq \det (\mathcal{F}(1)) = \det(\mathcal{F}) \otimes \mathcal{O}_X(2). $$