Moduli spaces of quasitrivial sheaves on the three dimensional projective space

Douglas Guimarães, Marcos Jardim

Published 2021-08-03Version 1

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