My problem involves holomorphic vector bundles $E,F$ of the same rank on $T\mathbb{C}P^1$. I have a short exact sequence of sheaves $$0\rightarrow E\rightarrow F\rightarrow Q\rightarrow 0.$$
I want to know what information I can glean about this torsion, coherent sheaf $Q$. In my case, I know a prioi that $Q$ is supported on a compact curve $D$, which is an effective divisor. However, I do not know if the curve is smooth. I've tried looking in Friedman's "Algebraic surfaces and holomorphic vector bundles", especially at elementary modifications, but I do not know if $Q$ is a vector bundle on $D$.
I believe that if the curve were smooth, then I could say that $Q\mid D$ is torsion-free away from a finite number of points (as $D$ is compact). But I'm not sure about what would happen if $D$ were not smooth.