Planar rank-one sheaves on $\mathbb{P}^3$, obstruction bundles, and divisor-supported Donaldson--Thomas series

Let $X=\mathbb{P}^3$ and let \[ α_n=(0,1,-\tfrac12,\tfrac16-n)\in H^{\mathrm{even}}(X,\mathbb{Q}) \] with respect to the basis $1,H,H^2,H^3$, where $H=c_1(\mathcal{O}_X(1))$. We prove that every Gieseker semistable sheaf on $X$ with Chern character $α_n$ is stable and uniquely of the form $ι_{P*}\ma