Let $X$ be a proper(say, smooth) variety and $E,F$ are coherent sheaves on it. Extensions of $E$ by $F$ are parametrised by a finite-dimensional vector space $\mathrm{Ext}^1(E,F)$. I am intersted in the isomorphism classes of sheaves which can be realized as such extensions.

Q1: Is it true, that the set $S=\{G\in Coh(X)|$there exists a short exact sequence$0\to F\to G\to E\to 0\}$ is finite?

Q2: Assuming the positive answer to Q1, is it true that subsets $S_G=\{e\in\mathrm{Ext}^1(E,F)|$middle sheaf of $e$ is $G$ $\}\subset \mathrm{Ext}^1(E,F)$ give a stratification on this affine space? In other words, can $S$ be ordered so that function, assigning to an $\mathrm{Ext}$ class the isomorphism class of the middle sheaf is semi-continous in Zariski topology.

I have checked this explicitly for $E=\mathcal{O}(2),F=\mathcal{O}(-2)$ on $\mathbb{P}^1$. There we get $\dim\mathrm{Ext}^1=3$ and $$S_{\mathcal{O}(2)\oplus\mathcal{O}(-2)}=0,S_{\mathcal{O}(1)\oplus\mathcal{O}(-1)} \text{ is a quadratic cone without the origin}, S_{\mathcal{O}\oplus\mathcal{O}}\text{ is the complement to this cone}$$

It seems that for any two line bundles on projective line the answer to $Q2$ is also positive since the splitting type of the extension is controlled by the rank of the connecting homomorphism in the $\mathrm{Ext}$'s exact sequence which is semi-continous.

Q3: Even if, in general, my assertions fail, what can be said about the line bundles on $\mathbb{P}^1$ case? I feel that this stratification might be connected to representation therory, as in the above example we get nilpotent cone of $\mathfrak{sl}_2$(though, this may be farfetched).

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    $\begingroup$ Q1 is false. Take a smooth projective curve $C$ of genus $\geq 2$, and fix a line bundle $L$ on $C$ of degree $>2g-2$. Any sufficiently general rank 2 vector bundle $E$ on $C$ with $\det E=L$ has a section which does not vanish at any point. This gives rise to an extension $0\rightarrow \mathcal{O}_C\rightarrow E\rightarrow L\rightarrow 0$. $\endgroup$ – abx Nov 3 '16 at 5:04

As abx explained, the answer to Q1 is negative (I had the same example in mind). Also this shows that the answer to Q2 is negative as well. In linw bundles on $P^1$ case, of course, the stratification is $SL_2$-invariant (and is easy to describe), but it is more coarse than the orbit stratification. Indeed, if the line bundles are $O$ and $O(-d)$, so that $Ext^1 = S^{d-2}(k^2)$, there are only $\lfloor d/2 \rfloor + 1$ isomorphism classes of extension sheaves, while the number of types of orbits is the namber of partitions of $d-2$, and moreover, when $d -2 \ge 4$ there are infinite families of orbits.


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