cut by the equation $\beta\circ\alpha=0$. One has the associated complex $$\xymatrix{
    M(\alpha,\beta): & {{\mathcal{U}}_{\vec{k}}}\ar[r]^-\alpha & {{\mathcal{V}}_{\vec{k}}}\ar[r]^-\beta & {{\mathcal{W}}_{\vec{k}}}\,.
    }$$

-   We define the quasi-affine variety ${L_{\vec{k}}}$ as the open subset of $\overline{L}_{\vec{k}}$ characterized by the following four conditions:

    -   the sheaf morphism $\alpha$ is a monomorphism;

    -   the sheaf morphism $\beta$ is an epimorphism;

    -   the vector space morphisms $\alpha\otimes k(y)$ have maximal rank for all closed points $y\in{\ell_\infty}$;

    -   if we consider the display associated with the monad $M(\alpha,\beta)$ as in eq. , and we restrict it to ${\ell_\infty}$, after twisting by ${\mathcal{O}}_{{\ell_\infty}}(-1)$ and taking cohomology, we get a vector space morphism $\Phi:=H^0\left(\nu|_{{\ell_\infty}}(-1)\right):H^0\left({\mathcal{A}}|_{{\ell_\infty}}(-1)\right)\longrightarrow H^0\left({{\mathcal{W}}_{\vec{k},\infty}}(-1)\right)$; note that $h^0\left({\mathcal{A}}|_{{\ell_\infty}}(-1)\right)= h^0\left({{\mathcal{W}}_{\vec{k},\infty}}(-1)\right)=nk_{3}$. We require that $$\det\Phi\neq0\,.$$

    -   the cohomology ${\mathcal{E}}_{\alpha,\beta}$ of the monad $M(\alpha,\beta)$ is torsion-free.

    We shall prove in Lemma \[6lm10\] that ${L_{\vec{k}}}$ is a smooth variety. We let ${\mathcal{E}}_{\alpha,\beta,\infty}=\left.{\mathcal{E}}_{\alpha,\beta}\right|_{{\ell_\infty}}$.

-   We consider the algebraic group $${G_{\vec{k}}}={\operatorname{Aut}}\left({{\mathcal{U}}_{\vec{k}}}\right)\times{\operatorname{Aut}}\left({{\mathcal{V}}_{\vec{k}}}\right)\times{\operatorname{Aut}}\left({{\mathcal{W}}_{\vec{k}}}\right).$$ This group acts naturally on ${L_{\vec{k}}}$ according to the following formulas: $$\left\{
    \begin{array}{rcl}
    \alpha & \mapsto & \alpha'=\psi\alpha\phi^{-1}\\
    \beta & \mapsto & \beta'=\chi\beta\psi^{-1}\\
    \end{array}
    \right.\qquad{\bar{\psi}}=(\phi,\psi,\chi)\in{G_{\vec{k}}}\,.
    \label{eqrho0}$$ This action will be called $\rho_0:{L_{\vec{k}}}\times{G_{\vec{k}}}\to{L_{\vec{k}}}$.

-   We introduce (see subsection \[pagPk\]) a principal ${\operatorname{GL}}(r,{\mathbb{C}})$ bundle ${P_{\vec{k}}}\stackrel{\tau}{\longrightarrow}{L_{\vec{k}}}$ whose fibre over $(\alpha,\beta)$ is naturally identified with the space of framings 