\label{eqMS}$$ where the locally-free sheaves ${\mathfrak{U}}$ and ${\mathfrak{W}}$ are $$\begin{cases}
\begin{aligned}
{\mathfrak{U}}&={\mathcal{O}}_T(0,-1)\otimes t^{\ast}_2R^1t_{2*}\left[{\mathfrak{F}}(-2,n-1)\right]\\
{\mathfrak{W}}&={\mathcal{O}}_T(1,0)\otimes t^{\ast}_2R^1t_{2*}\left[{\mathfrak{F}}(-1,0)\right].
\end{aligned}
\end{cases}$$ The locally-free sheaf ${\mathfrak{V}}$ is defined as an extension $$\begin{gathered}
\xymatrix{
0 \ar[r] & {\mathfrak{V}}_- \ar[r]^{{\mathfrak{c}}} & {\mathfrak{V}}\ar[r]^{{\mathfrak{d}}} & {\mathfrak{V}}_+ \ar[r] & 0\,,
}
\label{eqES}
\\
\text{where}\qquad
\begin{cases}
\begin{aligned}
{\mathfrak{V}}_+&:=t^{\ast}_2R^1t_{2*}\left[{\mathfrak{F}}(-2,n)\right]\\
{\mathfrak{V}}_-&:={\mathcal{O}}_T(1,-1)\otimes t^{\ast}_2R^1t_{2*}\left[{\mathfrak{F}}(-1,-1)\right]
\end{aligned}
\end{cases}
\notag\end{gathered}$$ and the morphisms ${\mathfrak{c}}$ and ${\mathfrak{d}}$ are canonically determined by ${\mathfrak{F}}$.

Moreover the association ${\mathfrak{F}}\mapsto M({\mathfrak{F}})$ is covariantly functorial.

To prove this Proposition, one needs the following result.

\[lm3\] Let $\mathcal{E}$ be a torsion-free sheaf on $\Sigma_n$ trivial at infinity. One has $$\begin{aligned}
 H^0(\mathcal{E}(p,q))=0\qquad&\text{\emph{for}}\qquad np+q\leq -1\,,
\\
 H^2(\mathcal{E}(p,q))=0\qquad&\text{\emph{for}}\qquad np+q\geq -(n+1)\,.
$$

When $\mathcal{E}$ is locally free, the proof is essentially the same as in [@Ki p. 24]. Otherwise we get the thesis by using the injection ${\mathcal{E}}\rightarrowtail {\mathcal{E}}^{**}$.

Let us consider the product scheme $\Sigma_n\times\Sigma_n\times S$, together with the canonical projections $p_{12}$, $p_{13}$ and $p_{23}$. Buchdahl [@Bu] proved the existence of a three-term locally-free resolution ${\mathcal{G}}^\bullet \twoheadrightarrow {\mathcal{O}}_\Delta$ of the structure sheaf ${\mathcal{O}}_{\Delta}$ of the diagonal $\Delta\subseteq \Sigma_n\times\Sigma_n$. This is given by $$\left\{\begin{array}{rcl} {\mathcal{G}}^0 &=& {\mathcal{O}}_{\Sigma_n\times\Sigma_n} \\[3pt]
{\mathcal{G}}^{-1} &=& \mathcal R^\ast \\[3pt]
{\mathcal{G}}^{-2} & = & {\mathcal{O}}_{\Sigma_n}(-1,-1) \boxtimes {\mathcal{O}}_{\Sigma_n}(-1,n-1)
\end{array}\right.\ ;$$ here $\mathcal R$ is the extension $$0 