Let $X$ be a smooth projective irreducible curve over an algebraically closed field $k$. Let $\mathcal E$ be a vector bundle (say, of rank $n$), and let $\mathcal F$ be another vector bundle of rank $n$ which is a degeneration of $\mathcal E$ (i.e. there exists a d.v.r. $R$ with residue field $k$ and fraction field $K$ and a family of vector bundles on $X$ over $Spec(R)$ whose fiber the special point of $Spec(R)$ is $\mathcal F$ and whose fiber over the generic point of $Spec(R)$ is the pull-back of $\mathcal E$ to $X\times Spec(K)$). One way to produce such a degeneration is to introduce a filtration on $\mathcal E$ whose associated graded is isomorphic to $\mathcal F$. My question is whether this is always the case, at least after some assumptions. More precisely, the question is this:
- Is $\mathcal F$ always isomorphic to the associated graded of $\mathcal E$ with respect to some filtration?
- If not, is it true under the assumption that $\mathcal E$ is stable?