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

Abstract

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 ${\alpha }_n$ is stable and uniquely of the form ${\iota }_{P\ast }{\mathcal{I}}_Z$ for a plane $P\subset X$ and a length-$n$ subscheme $Z\subset P$. Hence the coarse stable-sheaf moduli space is the relative Hilbert scheme of points on the universal plane over the dual projective space $X^{\vee }$. 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 \[ \sum_{n\ge 0} Γ_n q^n=\prod_{m\ge 1}(1-q^m)^{-7} \] for the natural point-inserted two-dimensional Donaldson--Thomas invariants on ${\mathbb{P}}^3$. 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 satisfying $H^1(D,{\mathcal{O}}_D)=0$, the divisor-supported moduli component is a Hilbert scheme of points on $D$, its obstruction bundle is the twisted tangent bundle ${\mathsf{T}}_D^{[n]}(N_{D/X})$, 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 ${\mathrm{Bl}}_p{\mathbb{P}}^3$ and of a rigid section in a Fano ${\mathbb{P}}^1$-bundle over ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

arXiv comment27 pages