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).