Let $S$ be a smooth projective surface and $C$ a smooth, irreducible curve contained in $S$. Let $E\_1$ and $E\_2$ be two vector bundles on $S$ having the same rank and assume they lie in a short exact sequence 
$$
0\to E\_1\stackrel{f}{\to} E\_2\to A\to 0, $$
with $A\in\mathrm{Pic}(C)$. In this situation $E\_1$ is called an elementary transformation of $E\_2$. Now, the map $f$ induces an injection $\mathrm{det}(f):\mathrm{det}E\_1\to \mathrm{det}E\_2$, whose cokernel is again supported on $C$ and coincides with $\mathrm{det}E\_2\otimes\mathcal{O}\_C$. Is there any relation between $A$ and $\mathrm{det}E\_2\otimes\mathcal{O}\_C$ (e.g, a map from one to the other one)?



Elementary transformations and determinant maps.
There is no relation. In fact, if $E\_2$ is fixed then $\det E\_2\otimes O\_C$ is fixed, while $A$ can be taken to be any invertible quotient of $E\_{2|C}$.


