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

Let $X=\PP^3$ and let   \[   \alpha_n=(0,1,-\tfrac12,\tfrac16-n)\in H^{\mathrm{even}}(X,\Q)   \]   with respect to the basis $1,H,H^2,H^3$, where $H=c_1(\OO_X(1))$.   We prove that every Gieseker semistable sheaf on $X$ with Chern character $\alpha_n$ is stable and uniquely of the form $\iota_{P*}\I_Z$ for a plane $P\subset X$ and a length-$n$ subscheme $Z\subset P$.   Hence the moduli space is the relative Hilbert scheme of points on the universal plane over the dual projective space $X^\dual$.   Using the standard perfect obstruction theory for stable sheaves on a Fano threefold, we identify the obstruction bundle with a relative Carlsson--Okounkov twisted tangent bundle and obtain a closed product formula   \[   \sum_{n\ge 0} \Gamma_n q^n=\prod_{m\ge 1}(1-q^m)^{-7}   \]   for the natural point-inserted two-dimensional Donaldson--Thomas invariants on $\PP^3$.   We then develop the analogous divisor-supported theory for a smooth divisor $D$ in a smooth projective Fano threefold, distinguishing moving and rigid divisors.   For a rigid divisor the divisor-supported moduli component is a Hilbert scheme of points on $D$, its obstruction bundle is the twisted tangent bundle $\Ttw_D^{[n]}(N_{D/X})$, and its generating series is   \[   \prod_{m\ge 1}(1-q^m)^{-(c_2(X)\cdot D+D^3)}.   \]   We work out the examples of the exceptional divisor in $\operatorname{Bl}_p\PP^3$ and of a rigid section in a Fano $\PP^1$-bundle over $\PP^1\times\PP^1$.