Suppose $L$ and $M$ are holomorphic line bundles over the same compact Riemann surface $X$. I want to study which extensions of $M$ by $L$, i.e., vector bundles $E$ in a short exact sequence of the form
$$ 0 \rightarrow L \rightarrow E \rightarrow M \rightarrow 0 $$
are unstable, in the sense of slope stability. I want extensions $E$ of degree zero (for now, but I don't think the degree is the most important thing here).
For example, assuming that $M$ has degree 1 and $L$ has degree -1, I was able to see that all non-trivial extensions (that are not the direct sum $L\oplus M$) are semistable, by doing some small calculations with degrees and bundles of homomorphisms. So the only unstable extension is the trivial one.
I am no longer able to do it if $L$ has degree -2 and $M$ has degree 2. The extensions are given by the cohomology group $H^1(M^{-1}L)$. The idea would be to get the unstable extensions as the kernel of some homomorphism from $H^1(M^{-1}L)$ to another cohomology group, probably provided by some short exact sequence of bundles giving a long exact sequence in cohomology.
Sadly I am not yet used to this enough to see what to do, but it seems that, in the case of line bundles and rank 2 extensions, the more basic stuff should be enough.
Any suggestions? If you have some book suggestions, I would greatly appreciate that too.