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

On the Hilbert scheme of the moduli space of torsion free sheaves on surfaces