T$, the stalk ${\mathcal{E}}_{(x,s)}$ is torsion-free. Let $\mathcal T$ be torsion submodule of ${\mathcal{E}}$, so that one has an exact sequence $ 0 \to \mathcal T \to {\mathcal{E}}\to {\mathcal{E}}^{\ast\ast}$. By localizing at a closed point $(x,s)\in T$, we obtain $ \mathcal T_{(x,s)}=0$. By Lemma 2.8 in [@Eis], one has $\mathcal T=0$.

The morphism of sheaves $$\label{dd}
{\operatorname{\mathcal{H}\!\textit{om}}}({\mathcal{E}},{\mathcal{E}}')\stackrel{f}{\to} {\operatorname{\mathcal{H}\!\textit{om}}}({\mathcal{E}}^{**},\left({\mathcal{E}}'\right)^{**})$$ obtained by taking double duals is injective as ${\mathcal{E}}'$ is torsion-free by Lemma \[torsionfree\]. Since for all closed $s\in S$ the sheaves ${\mathcal{E}}_s$ and ${\mathcal{E}}'_s$ are torsion-free, by appling Lemma \[pro1\] one deduces that also the restricted morphisms $f_s$ are injective for all closed $s\in S$. At the same time the sheaf ${\operatorname{\mathcal{H}\!\textit{om}}}\left({\mathcal{E}}^{**},\left({\mathcal{E}}'\right)^{**}\right)$ is flat on $S$ by its local freeness and the flatness of the projection $t_2$. From this we can prove that ${\operatorname{coker}}f$ is flat on $S$ by a reasoning similar to the proof of Lemma \[lmFlat\]. The thesis follows from [@Har Prop. III. 9.1A.(e)].

Statement of the main result {#Main result}
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Let $\Sigma_n$ be the $n$-th Hirzebruch surface, i.e., the projective closure of the total space of the line bundle $\mathcal{O}_{\mathbb{P}^1}(-n)$. We denote by $F$ the class in ${\operatorname{Pic}}(\Sigma_n)$ of the natural ruling $\Sigma_n\longrightarrow{\mathbb{P}^1}$, by $H$ the class 