Two infinite series of moduli spaces of rank 2 sheaves on ${\mathbb{P}}^3$

We describe new components of the Gieseker–Maruyama moduli scheme $\mathcal{M}(n)$ of semistable rank 2 sheaves $E$ on ${\mathbb{P}}^3$ with $c_1(E)=0$, $c_2(E)=n$ and $c_3(E)=0$ whose generic point corresponds to non locally free sheaves. We show that such components grow in number as $n$ grows, and discuss how they intersect the instanton component. As an application, we prove that $\mathcal{M}(2)$ is connected, and identify a connected subscheme of $\mathcal{M}(3)$ consisting of 7 irreducible components.