  The aim of this paper is to determine a bound of the dimension of an
irreducible component of the Hilbert scheme of the moduli space of torsion-free
sheaves on surfaces. Let $X$ be a non-singular irreducible complex surface and
let $E$ be a vector bundle of rank $n$ on $X$. We use the $m$-elementary
transformation of $E$ at a point $x \in X$ to show that there exists an
embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli
space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;c_1,c_2+m)$ which induces
an injective morphism from $X \times M_{X,H}(n;c_1,c_2)$ to $Hilb_{\,
\mathfrak{M}_{X,H}(n;c_1,c_2+m)}$.
