**I. General Question**


Consider a one-parameter family of vector bundles $E\_t$ on a smooth projective variety $X$ with fixed Chern character $v$. Suppose $E\_t$ is Gieseker stable when $t\neq 0$ and $E\_0$ is *not* Gieseker semi-stable. **Is there a way to find the limit $\lim\_{t\to 0}E\_t$ in the Gieseker semi-stable moduli space $\overline{\mathcal{M}}\_v(X)$?**


Typically, the limit is a semi-stable coherent sheaf and not necessarily locally free. I'm wondering if there is a way that allows us "modify" the family and produce the semi-stable sheaf at special fiber after a suitable base change?


A counterpart that I have in mind is the Stable Reduction theorem [[Harris-Morrison](https://link.springer.com/book/10.1007/b98867), Prop. 3.17] for curves, which provides us an algorithm to find the limiting stable curve for a flat family of curves with bad singularities at the special fiber. I'm not sure if "stable reduction" is too much to hope for in the realm of the vector bundles.


**II. An Example**


Here is an example I'm working on. On a smooth cubic threefold $X$, there is a rank-two vector bundle $E$ associated to an elliptic quintic curve $C$ on $X$ via Serre construction of the ideal sheaf $I\_C$. One can show $c\_1(E)=0$ and $c\_2(E)=2$. When $C$ is projectively normal, $E$ is Gieseker stable, while when $C$ is not projectively normal (equivalently, $C$ is contained in a hyperplane), $E$ is *not* Gieseker semi-stable (in fact, it is slope semi-stable, though). See [[Markushevich and Tikhomirov](https://arxiv.org/pdf/math/9809140.pdf), Prop. 2.6]. Therefore, a smooth family $C\_t$ of elliptic quintics on $X$ with $C\_0$ contained in a hyperplane will produce a family of vector bundles $E\_t$ in question.


So how to find the limit $\lim\_{t\to 0}E\_t$ in $\overline{\mathcal{M}}\_{2,0,2}(X)$?


[[Markushevich and Tikhomirov](https://arxiv.org/pdf/math/9809140.pdf), Prop. 2.5] indicates there is a unique pair of lines $L\_1$ and $L\_2$ on $X$ associated to the non projectively normal curve $C\_0$. On the other hand, [[Druel](https://arxiv.org/abs/math/0002058), Theorem 3.5] and [[Beauville](https://arxiv.org/abs/math/0005017), Prop. 6.2(a)] indicate the strict semi-stable sheaves have the form $I\_{M\_1}\oplus I\_{M\_2}$ for a pair of lines $M\_1,M\_2$ on $X$.


These results seem to suggest to us what the limit sheaf is. However, I still cannot find a geometric family of sheaves in $\overline{\mathcal{M}}\_{2,0,2}(X)$ with $I\_{L\_1}\oplus I\_{L\_2}$ appears at time 0.


I appreciate it if anyone could help. Happy new year!



Is there a stable reduction for a family of vector bundles?
Consider the Harder-Narasimhan filtration of $E\_0$. Assume for simplicity it has length 2:
$$
0 \to F\_0 \to E\_0 \to E\_0/F\_0 \to 0,
$$
where $F\_0$ and $E\_0/F\_0$ are semistable and the slope of $F\_0$ is bigger than the slope of $E\_0/F\_0$. If $B$ is the parameter space of your family and $i \colon X \to X \times B$ is the embedding of the central fiber, consider the sheaf $E'$ on $X \times B$ defined by the exact sequence
$$
0 \to E' \to E \to i\_\*(E\_0/F\_0) \to 0.
$$
Note that the new family $E'$ has $E'\_t = E\_t$ for $t \ne 0$. On the other hand, if you restrict this sequence to the central fiber and take into account the isomorphism $L\_1i^\*(i\_\*G) \cong G$ (because the normal bundle of the central fiber is trivial of rank 1), you get an exact sequence
$$
0 \to E\_0/F\_0 \to i^\*E' \to E\_0 \to E\_0/F\_0 \to 0,
$$
which implies that $E'$ is flat over $B$ and $E'\_0$ fits into an exact sequence
$$
0 \to E\_0/F\_0 \to E'\_0 \to F\_0 \to 0.
$$
Now the sheaf $E'\_0$ is "more semistable" than $E\_0$, so iterating this procedure you can eventually modify your family to the one with semistable central fiber.


