Abstract

A torsion-free sheaf $E$ on a projective variety $X$ is called quasitrivial 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 quasitrivial sheaves on $X$ with $E^{\vee \vee }∕E$ being a 0-dimensional sheaf of length $n$ via the Quot scheme of points $\mathrm{Quot}<!--FUNCTION\; APPLICATION-->({\mathcal{𝒪}}_X^{\oplus r},n)$. We show that when $X$ is a suitable projective variety, ${\mathcal{𝒩}}_X(r,n)$ is empty for $r>n$, while ${\mathcal{𝒩}}_X(n,n)$ has no stable points and is isomorphic to the symmetric product ${\mathrm{Sym}<!--FUNCTION\; APPLICATION-->}^n(X)$. Our main result is the construction of an irreducible component of ${\mathcal{𝒩}}_X(r,n)$ of dimension $n(d+r-1)-r^2+1$, where $d=\dim <!--FUNCTION\; APPLICATION-->(X)$ when $r<n$. Furthermore, if we restrict to $X={\mathbb{P}}^3$, this is the only irreducible component when $n\le 10$.

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