Moduli spaces of quasi-trivial sheaves
Douglas Guimarães, Marcos Jardim
TL;DR
This work studies the Gieseker--Maruyama moduli space \(\mathcal{N}_X(r,n)\) of semistable quasi-trivial sheaves with \(E^{\vee\vee} \cong \mathcal{O}_X^{\oplus r}\) on a polarized projective variety \((X,A)\), connecting it to Quot schemes of points. It proves a complete dichotomy: the space is empty for \(r>n\), while \(\mathcal{N}_X(n,n) \cong \mathrm{Sym}^n(X)\) with no stable points; for \(r<n\) there exists an irreducible component of dimension \(n(d+r-1)-r^2+1\). The authors develop a framework linking semistability to GIT stability via \(\mathrm{GL}_r\) actions on Quot schemes and construct irreducible components explicitly, notably for rank \(2\) and then inductively for higher ranks. In the special case \(X=\mathbb{P}^3\), they show irreducibility for \(n\le 10\), leveraging known irreducibility of the affine Quot scheme and relating the moduli to the variety of commuting matrices. These results illuminate the geometry of quasi-trivial sheaves and their moduli, and connect them to Hilbert/Quot schemes and representation-theoretic data.
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 $μ$-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 $n\le10$.