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

Abstract

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 ∈ 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,H (n; c 1 , c 2 + m) which induces an injective morphism from X × M X,H (n; c 1 , c 2 ) to Hilb MX,H (n;c1,c2+m) .