Moduli spaces of quasi-trivial sheaves

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 $Sy{m}^n(X)$. 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 $n\le 10$.

References in corpus (3)

• Hilbert schemes of lines and conics and automorphism groups of Fano threefolds
• Two infinite series of moduli spaces of rank 2 sheaves on ${\mathbb{P}}^3$
• Irreducible components of the moduli space of rank 2 sheaves of odd determinant on the projective space