Let X be a projective scheme over a field k , and

$0\rightarrow F'\rightarrow F \rightarrow F''\rightarrow 0$ be a short exact sequence of semi-stable sheaves with same reduced Hilbert polynomial.

I want to find a flat family $E$ of semi stable sheaves parametrized by affine line $A^1$ such that

$E_0 \simeq F'\bigoplus F''$

$E_t\simeq F $ for all $t\in A^1\setminus \lbrace 0\rbrace$

and this problem is in Lemma 8.3.4 in this PDF

In this PDF,what is

"the coherent sheaf over X × $A^1$ given by this extension class."?


You can construct a "universal (or tautological)" family of extensions $\mathcal{E}$ on $X\times Ext^1(F'',F')$: $0\rightarrow p_1^{*}F'\rightarrow \mathcal{E} \rightarrow p_1^{*}F''\rightarrow 0$, with $p_1: X\times Ext^1(F'',F')\rightarrow X$, such that for every $e\in Ext^1(F'',F')$ the restriction $\mathcal{E}_e$ is just $0\rightarrow F'\rightarrow \mathcal{E}_e\rightarrow F'' \rightarrow 0$, the extension of $F'$ by $F''$ given by $e$.

Now you take $e\in Ext^1(F'',F')$ and look at the $\mathbb{A}^1$ inside $Ext^1(F'',F')$ defined by the line through $e$ and just restrict $\mathcal{E}$ to $X\times\mathbb{A}^1$.

For the construction of $\mathcal{E}$ see the article "Universal families of extensions" by H.Lange.


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