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 $0$?
1 Answer
$\begingroup$
$\endgroup$
3
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 $$\mu(W)=\frac{1}{2} > \frac{1}{3} = \mu(V).$$
-
$\begingroup$ This is the only example in the sense that V can not be a non-split extension of rank 2 degree 1 bundle and a rank 1 degree 0 bundle. Also the SCSS subbundle could only be a rank 2 degree 1 bundle. $\endgroup$– user100841Jul 2, 2018 at 10:39
-
$\begingroup$ Maybe you mean that this is the only example on an elliptic curve? $\endgroup$ Jul 2, 2018 at 10:48
-
$\begingroup$ No i mean for all curves it can only be such a split extension. This is becasuse in my hypothesis, the rank 2 degree 1 bundle is stable. $\endgroup$– user100841Jul 2, 2018 at 10:53