I am reading "Lectures on vector bundles" by Le Potier and am confused about a statement in the proof of the existence theorem on page 144, after Lemma 8.6.6.
Let $X$ be a projective curve (can assume non-singular) and $S$ be an affine algebraic variety. Fix integers $r,d$ and $E$ a free bundle on $X \times S$ of rank $r$. Denote by $\mathrm{Hilb}^{r,d}(E/S)$ the relative Hilbert scheme parametrizing quotients of $E$ of degree $d$ and rank $r$ (see page 142 in the literature for details of construction).
For any closed point $q \in \mathrm{Hilb}^{r,d}(E/S)$ corresponding to an exact sequence $$0 \to G \to E(q) \to F \to 0$$, we have an exact sequence: $$0 \to \mathrm{Hom}(G,F) \to T_q\mathrm{Hilb}^{r,d}(E/S) \to T_sS \xrightarrow{\omega} \mathrm{Ext}^1(G,F)$$
Suppose that $S$ is non-singular at the point $s$. Then, the part following Lemma 8.6.6 seems to suggest that the map $\omega$ is surjective. I do not understand how this follows? Am I missing something? Any hint/suggestion will be most welcome.
The long exact sequence associated to $0\rightarrow G(s)\rightarrow E\rightarrow F(s)\rightarrow 0$ after applying $Hom(-,F(s))$ gives
$\ldots\rightarrow Hom(G(s),F(s))\rightarrow Ext^1(F(s),F(s))\rightarrow Ext^1(E,F(s))\rightarrow \ldots$
But since $E=H\otimes \mathcal{O}_X$ we see that $Ext^1(E,F(s))=0$ since $H^1(F(s))=0$ by the remarks before Lemma 8.6.6.
So the map in question is indeed surjective.