that $S$ is affine, and that the sheaves $R^1t_{2*}\left({\mathfrak{F}}(p,q)\right)$ are trivial for $(p,q)\in{\mathscr{I}}$. There are isomorphisms $${\mathfrak{U}}\simeq t_1^*{{\mathcal{U}}_{\vec{k}}}\,;\qquad
{\mathfrak{V}}\simeq t_1^*{{\mathcal{V}}_{\vec{k}}}\,;\qquad
{\mathfrak{W}}\simeq t_1^*{{\mathcal{W}}_{\vec{k}}}\,,$$ the sheaves ${{\mathcal{U}}_{\vec{k}}}$, ${{\mathcal{V}}_{\vec{k}}}$ and ${{\mathcal{W}}_{\vec{k}}}$ being defined as in eq. .

A trivialization for $R^1t_{2*}\left({\mathfrak{F}}(p,q)\right)$ amounts to choosing a closed point $s_0\in S$ and an isomorphism $$R^1t_{2*}\left({\mathfrak{F}}(p,q)\right)\stackrel{\sim}{\longrightarrow}{\mathcal{O}}_S\otimes\left[R^1t_{2*}\left({\mathfrak{F}}(p,q)\right)\otimes k(s_0)\right].$$ Since ${\mathfrak{F}}_{s_0}$ is torsion-free and trivial at infinity, from Lemma \[lm3\] and from the Semicontinuity Theorem one obtains the isomorphism $$R^1t_{2*}\left({\mathfrak{F}}(p,q)\right)\otimes k(s_0)\simeq H^1\left({\mathfrak{F}}_{s_0}(p,q)\right)\,.$$ The dimensions of the vector spaces $H^1({\mathfrak{F}}_{s_0}(p,q))$ can be computed by means of Riemann-Roch Theorem and Lemma \[lm3\]: $$h^1\left({\mathfrak{F}}_{s_0}(p,q)\right)=
\begin{cases}
k_1 & \text{for $(p,q)=(-2,n-1)$}\\
k_2 & \text{for $(p,q)=(-1,-1)$}\\
k_3 & \text{for $(p,q)=(-1,0)$}\\
k_4 & \text{for $(p,q)=(-2,n)$}\\
\end{cases}$$ where $k_i$, $i=1,\ldots,4$ are as in eq. . As a consequence the sheaves $R^1t_{2*}\left({\mathfrak{F}}(-2,n-1)\right)$, $R^1t_{2*}\left({\mathfrak{F}}(-1,-1)\right)$, $R^1t_{2*}\left({\mathfrak{F}}(-1,0)\right)$ and $R^1t_{2*}\left({\mathfrak{F}}(-2,n)\right)$ are free of ranks $k_1,\dots,k_4$ respectively, so that $${\mathfrak{U}}\simeq
{\mathcal{O}}_T(0,-1)^{\oplus k_1}\,;\qquad
{\mathfrak{W}}\simeq
{\mathcal{O}}_T(1,0)^{\oplus k_3}\,;\qquad
{\mathfrak{V}}_-
\simeq
{\mathcal{O}}_T^{\oplus k_4}\,;\qquad
{\mathfrak{V}}_+
\simeq
{\mathcal{O}}_T(1,-1)^{\oplus k_2}\,.$$ The thesis follows for ${\mathfrak{U}}$ and ${\mathfrak{W}}$. By plugging ${\mathfrak{V}}_-$ and ${\mathfrak{V}}_+$ into the sequence eq. , the latter splits, since $$\begin{aligned}
{\operatorname{Ext}}^1\left({\mathcal{O}}_T(1,-1),{\mathcal{O}}_T\right)&\simeq H^1\left(T,{\mathcal{O}}_T(-1,1)\right)\simeq\\
\tag{$*$} &\simeq H^0\left(S,R^1t_{2*}{\mathcal{O}}_T(-1,1)\right)\simeq\\
&\simeq H^0\left(S,R^1t_{2*}\left[t_1^*{{\mathcal{O}}_{\Sigma_n}}(-1,1)\right]\right)\simeq\\
&\simeq H^0(S,{\mathcal{O}}_S)\otimes H^1\left(\Sigma_n,{{\mathcal{O}}_{\Sigma_n}}(-1,1)\right)=0\end{aligned}$$ (the isomorphism ($*$) holds true as $S$ is affine and the vanishing is a consequence of Lemma 