\mathcal{E}\stackrel{\sim}{\longrightarrow}\mathcal{E}'$ such that the following diagram commutes: $$\xymatrix{
{\mathcal{E}}|_{\ell_{\infty}}\ar[r]^{\theta}\ar[d]_{\Lambda_\infty} & {\mathcal{O}}_{{\ell_\infty}}^{\oplus r} \\
{\mathcal{E}}'|_{\ell_{\infty}}\ar[ru]_{\theta'} &  }
\label{2eq14}$$ where $\Lambda_\infty:=\Lambda|_{\ell_{\infty}} $.

Let ${{\mathcal{M}}^n(r,a,c)}$ be the set of isomorphism classes of framed sheaves on $\Sigma_n$ having rank $r$, first Chern class $aE$, and second Chern class $c$: we shall prove that this set can be endowed with a structure of a smooth algebraic variety. We restrict ourselves to the case $n\geq1$ and assume that the framed sheaves are normalized in such a way that $0\leq a\leq r-1 $.

In order to simplify the statement of Theorem \[teo2\] we introduce some notation.

-   We denote by $\vec{k}$ a quadruple $(n,r,a,c)$, and define $k_i\in{\mathbb{Z}}$, $i=1,\ldots,4$ as follows: $$k_1=c+\dfrac{1}{2}na(a-1)\qquad\text{and}\qquad
    \begin{cases}
    \begin{aligned}
    k_2&=k_1+na\\
    k_3&=k_1+(n-1)a\\
    k_4&=k_1+r-a\,.
    \end{aligned}
    \end{cases}
    \label{eqki}$$ The other definitions needed to state Theorem \[teo2\] makes sense only under the assumpition $k_1\geq0$. The Theorem itself will give a deeper meaning to this inequality.

-   We introduce the locally-free sheaves $$\left\{
    \begin{aligned}
    {{\mathcal{U}}_{\vec{k}}}&:={{\mathcal{O}}_{\Sigma_n}}(0,-1)^{\oplus k_1}\\
    {{\mathcal{V}}_{\vec{k}}}&:={{\mathcal{O}}_{\Sigma_n}}(1,-1)^{\oplus k_2} \oplus {{\mathcal{O}}_{\Sigma_n}}^{\oplus k_4}\\
    {{\mathcal{W}}_{\vec{k}}}&:={{\mathcal{O}}_{\Sigma_n}}(1,0)^{\oplus k_3}.
    \end{aligned}
    \right.
    \label{eqUVWk}$$ We shall write ${{\mathcal{U}}_{\vec{k},\infty}}$ in place of $\left.{{\mathcal{U}}_{\vec{k}}}\right|_{{\ell_\infty}}$, etc.

-   We introduce the vector space $${\mathbb{V}_{\vec{k}}}:={\operatorname{Hom}}\left({{\mathcal{U}}_{\vec{k}}},{{\mathcal{V}}_{\vec{k}}}\right)\oplus{\operatorname{Hom}}\left({{\mathcal{V}}_{\vec{k}}},{{\mathcal{W}}_{\vec{k}}}\right)\,,$$ whose elements will be denoted by $(\alpha,\beta)$.

-   Let $\overline{L}_{\vec{k}}$ be the affine subvariety of ${\mathbb{V}_{\vec{k}}}$ 