of the section of the ruling squaring to $n$, and by $E$ the class of the section squaring to $-n$. One has $E=H-nF$ and $F^2=0$, $H\cdot F=1$. We fix a curve $\ell_{\infty}\simeq{\mathbb{P}^1}$ in $\Sigma_n$ linearly equivalent to $H$ and call it the “line at infinity”.

One has ${\operatorname{Pic}}(\Sigma_n)={\mathbb{Z}}H\oplus{\mathbb{Z}}F$. For any sheaf ${\mathcal{E}}$ of ${{\mathcal{O}}_{\Sigma_n}}$-modules we shall write $$\mathcal{E}(p,q):=\mathcal{E}\otimes{{\mathcal{O}}_{\Sigma_n}}(pH+qF)\qquad p,q\in \mathbb{Z}\,.$$

\[lmVanH\] $$\begin{aligned}
H^0({{\mathcal{O}}_{\Sigma_n}}(p,q))\neq0\qquad&\text{if and only if}\qquad
\begin{cases}
p\geq0\\
np+q\geq0\,;
\end{cases}\\
H^1({{\mathcal{O}}_{\Sigma_n}}(p,q))\neq0\qquad&\text{if and only if}\qquad
\begin{cases}
p\geq0\\
q\leq-2
\end{cases}
\qquad\text{or}\qquad
\begin{cases}
p\leq-2\\
q\geq n\,;
\end{cases}\\
H^2({{\mathcal{O}}_{\Sigma_n}}(p,q))\neq0\qquad&\text{if and only if}\qquad
\begin{cases}
p\leq-2\\
np+q\leq-(n+2)\,.
\end{cases}\end{aligned}$$

Similar to King’s proof for the case $n=1$ [@Ki pp. 22-23].

Among the several definitions of framed sheaves available in the literature, we shall adopt the following.

\[DEf2\] A *framed sheaf* is a pair $(\mathcal{E},\theta) $, where

1.  $\mathcal{E}$ is a torsion-free sheaf on $\Sigma_n$ such that $$\mathcal{E}|_{\ell_{\infty}}\simeq{\mathcal{O}}_{{\ell_\infty}}^{\oplus r},
    \label{6eq34}$$ with $r={\operatorname{rk}}({\mathcal{E}})$.

2.  $\theta $ is a fixed isomorphism $\theta:\mathcal{E}|_{\ell_{\infty}}\stackrel{\sim}{\longrightarrow}{\mathcal{O}}_{{\ell_\infty}}^{\oplus r}$;

Condition implies $c_1({\mathcal{E}})\propto E $. The isomorphism $\theta$ is the so-called *framing at infinity*. By “sheaf trivial at infinity” we shall mean a sheaf satisfying condition (without any assigned framing).

\[DEf3\] An isomorphism $\Lambda$ between two framed sheaves $(\mathcal{E},\theta) $ and $(\mathcal{E}',\theta') $ is an isomorphism $\Lambda: 