Moduli spaces of quasi-trivial sheaves

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 $n\le10$.