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

TL;DR

This paper establishes bounds on the dimension of irreducible components of the Hilbert scheme associated with the moduli space of torsion-free sheaves on complex surfaces, using elementary transformations and embeddings from Grassmannians.

Contribution

It introduces a method to embed Grassmannians into the moduli space, providing new bounds on the Hilbert scheme's dimension for torsion-free sheaves on surfaces.

Findings

Existence of an embedding from Grassmannians to the moduli space.

Injection from the product of the surface and moduli space to the Hilbert scheme.

Bound on the dimension of irreducible components of the Hilbert scheme.

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 $Hilb_{\, \mathfrak{M}_{X,H}(n;c_1,c_2+m)}$.