\to {\mathcal{O}}_{\Sigma_n}(1, 0) \boxtimes {\mathcal{O}}_{\Sigma_n}( 1,-n ) \to \mathcal R \to {\mathcal{O}}_{\Sigma_n}(0,1) \boxtimes {\mathcal{O}}_{\Sigma_n}(0,1) \to 0$$ corresponding to the image in $H^1( \Sigma_n\times\Sigma_n,  {\mathcal{O}}_{\Sigma_n}(1,-1) \boxtimes {\mathcal{O}}_{\Sigma_n}(1,-n-1) )$ of $t$ under the connecting homomorphism of the exact sequence $$\begin{gathered}
 0 \to  {\mathcal{O}}_{\Sigma_n}(1,-1) \boxtimes {\mathcal{O}}_{\Sigma_n}(1,-n-1) ) \to {\mathcal{O}}_{\Sigma_n}(1,0) \boxtimes {\mathcal{O}}_{\Sigma_n}(1,-n) \\
\to {\mathcal{O}}_{\Sigma_n}(1,0) \boxtimes {\mathcal{O}}_{\Sigma_n}(1,-n) \otimes {\mathcal{O}}_Y\to 0\,. $$\end{gathered}$$ Here $Y$ is a suitable divisor in $\Sigma_n\times\Sigma_n$, while $t$ is a section of $ {\mathcal{O}}_{\Sigma_n}(0,1) \boxtimes {\mathcal{O}}_{\Sigma_n}(0,-1) \otimes {\mathcal{O}}_Y$ whose zero locus is $\Delta$.

Given any sheaf ${\mathfrak{F}}$ on $T$ fullfilling condition $\vec{k}$, we introduce the complex $${\mathcal{C}}^\bullet=\xymatrix{
\mathcal{C}^{-2} \ar[r] & \mathcal{C}^{-1} \ar[r] & \mathcal{C}^{0}
}=\left(p_{12}^*{\mathcal{G}}^\bullet\right)\otimes\left[p_{23}^{\ast}({\mathfrak{F}}(-1,0))\right]\,.$$ There are two spectral sequences, both abutting to the hyperdirect image $\mathbb{R}^\bullet p_{13\ast}(\mathcal{C}^{\bullet})$. From the first spectral sequence one gets $$\mathbb{R}^ip_{13\ast}(\mathcal{C}^{\bullet})=
\begin{cases}
{\mathfrak{F}}(-1,0)\qquad&\text{if}\;i=0\\
0\qquad&\text{otherwise}\,.
\end{cases}
\label{4eq37}$$ By using Lemma \[lm3\], one can obtain from the second exact sequence a complex that, when twisted by ${\mathcal{O}}_T(1,0)$, yields $M({\mathfrak{F}})$.

The functoriality follows from the construction.

This proof implies that the sheaves $R^1t_{2*}\left({\mathfrak{F}}(p,q)\right)$ are locally free for $$(p,q)\in{\mathscr{I}}=\{(-2,n-1),(-1,0),(-2,n),(-1,-1)\}\,.$$

The variety ${L_{\vec{k}}}$
---------------------------

As a straightforward consequence of Proposition \[teo1\] we get the following result.

\[corLocOnS\] Let the sheaf ${\mathfrak{F}}$ and the monad $M({\mathfrak{F}})$ be as in Proposition \[teo1\]. Assume 