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Suppose we have a non trivial short exact sequence, $$0\longrightarrow F'\longrightarrow F\longrightarrow F''\longrightarrow0,$$ where $F$, $F'$ and $F''$ are semistable sheaves with the same reduced Hilbert polinomial. How can I construct a flat family $\mathcal{F}$ of semistable sheaves parametrized by $\mathbb{A}^1$ such that $$\mathcal{F}_0\cong F'\oplus F'',\qquad\mathcal{F}_t\cong F\quad\forall\,t\neq0?$$ If it is important, I'm working on a projective scheme.

P.S. I'm also wondering if, given a semistable sheaf $F$, I can always find such a short exact sequence. Thank you!

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