On Extensions of locally free sheaves of rank $2$ on surface 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
 A: 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: 
  
  
*
  
*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.
  
*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 
$$
L \simeq \det (\mathcal{F}(1)) = \det(\mathcal{F}) \otimes \mathcal{O}_X(2).
$$
