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

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 nonsingular 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 is an element of X to show that there exists an embedding from the Grassmannian variety G(E-x, m) into the moduli space of torsion-free sheaves M-X,M-H(n; c(1), c(2) + m) which induces an injective morphism from X x M-X,M-H(n; c(1), c(2)) to Hilb(MX,H(n; c1, c2+m)).