Is there a curve (compact Riemann surface) $X$, a (holomorphic) vector bundle $\mathcal{E}$ over $X$, and an integer $d>0$ such that the following hold?
- There are surjections $\rho : \mathcal{E}\rightarrow \mathcal{L}$, $\rho ':\mathcal{E}\rightarrow \mathcal{L}'$ where $\mathcal{L},\mathcal{L}'$ are line bundles of the same degree $-d$, yet $\mathcal{L} \ncong \mathcal{L}'$.
- $\text{ker}(\rho)$ and $\text{ker}(\rho')$ are line bundles of degree $>-d$.