Is there a non-semistable bundle of rank $3$ and degree $1$ which is an extension of a stable bundle of rank $2$ degree $1$ by a stable bundle of rank $1$ degree $0$?



semistability of an extension of bundles
On an elliptic curve $C$, take $$V=E\_2(p) \oplus \mathcal{O}\_C,$$ where $p \in C$ is a point and $E\_2(p)$ is the unique non-split extension $$0 \to \mathcal{O}\_C \to E\_2(p) \to \mathcal{O}\_C(p) \to 0. $$


The rank $2$ subbundle $W=E\_2(p)$ strictly destabilizes $V$, because $$\mu(W)=\frac{1}{2} > \frac{1}{3} = \mu(V).$$


