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 $x\in X$ 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 $Hil{b}_{{\mathfrak{M}}_{X,H}(n;c_1,c_2+m)}$.

19 pages