**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][1], 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][2], 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][2], 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][3], Theorem 3.5] and [[Beauville][4], 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!

[1]: https://link.springer.com/book/10.1007/b98867
[2]: https://arxiv.org/pdf/math/9809140.pdf
[3]: https://arxiv.org/abs/math/0002058
[4]: https://arxiv.org/abs/math/0005017