Moduli spaces of quasi-trivial sheaves

TL;DR

This paper investigates the moduli space of quasi-trivial sheaves on projective varieties, revealing conditions for emptiness, structure, and irreducible components, especially on projective three-space.

Contribution

It constructs explicit irreducible components of the moduli space for quasi-trivial sheaves and analyzes their properties on various projective varieties.

Findings

The moduli space is empty when rank exceeds the length n.

When rank equals n, the moduli space is isomorphic to the symmetric product of X.

An irreducible component of specified dimension is constructed for r<n.

Abstract

A torsion-free sheaf $E$ on a projective variety $X$ is called quasi-trivial if $E^{\vee\vee}=\mathcal{O}_{X}^{\oplus r}$. While such sheaves are always $\mu$-semistable, they may not be semistable. We study the Gieseker--Maruyama moduli space $\mathcal{N}_X(r,n)$ of rank $r$ semistable quasi-trivial sheaves on $X$ with $E^{\vee\vee}/E$ being a 0-dimensional sheaf of length $n$ via the Quot scheme of points $Quot(\mathcal{O}_{X}^{\oplus r},n)$. We show that, when $(X,A)$ is a good projective variety, then $\mathcal{N}_X(r,n)$ is empty when $r>n$, while $\mathcal{N}_X(n,n)$ has no stable points and is isomorphic to the symmetric product $Sym^n(X)$. Our main result is the construction of an irreducible component of $\mathcal{N}_X(r,n)$ of dimension $n(d+r-1)-r^2+1$ when $r<n$. Furthermore, if we restrict to $X=\mathbb{P}^3$ this is the only irreducible component when $n\le10$.