Planar rank-one sheaves on ℙ3, obstruction bundles, and divisor-supported Donaldson--Thomas series
Abstract
Let X=ℙ3 and let αn=(0,1,−1/2,1/6−n)∈ Heven(X,ℚ) with respect to the basis 1,H,H2,H3, where H=c1(𝒪X(1)). We prove that every Gieseker semistable sheaf on X with Chern character αn is stable and uniquely of the form ιP*ℐZ for a plane P⊂ X and a length-n subscheme Z⊂ 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∨. 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 ∑n≥ 0 Γn qn=∏m≥ 1(1−qm)−7 for the natural point-inserted two-dimensional Donaldson--Thomas invariants on ℙ3. 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 H1(D,𝒪D)=0, the divisor-supported moduli component is a Hilbert scheme of points on D, its obstruction bundle is the twisted tangent bundle TD[n](ND/X), and its generating series is ∏m≥ 1(1−qm)−(c2(X)· D+D3). We work out the examples of the exceptional divisor in Blpℙ3 and of a rigid section in a Fano ℙ1-bundle over ℙ1×ℙ1.
Versions
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BibTeX
@misc{anderson2026planar,
title = {{Planar rank-one sheaves on $\mathbbP^3$, obstruction bundles, and divisor-supported Donaldson--Thomas series}},
author = {Reginald Anderson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28501},
eprint = {2609.28501},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28501},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Planar rank-one sheaves on ℙ^3, obstruction bundles, and divisor-supported Donaldson--Thomas series
AU - Reginald Anderson
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.28501
UR - https://arxiv.org/abs/2609.28501
AB - Let X=ℙ^3 and let α_n=(0,1,−1/2,1/6−n)∈ H^even(X,ℚ) with respect to the basis 1,H,H^2,H^3, where H=c_1(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*ℐ_Z for a plane P⊂ X and a length-n subscheme Z⊂ 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^∨. 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 ∑_n≥ 0 Γ_n q^n=∏_m≥ 1(1−q^m)^−7 for the natural point-inserted two-dimensional Donaldson--Thomas invariants on 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,𝒪_D)=0, the divisor-supported moduli component is a Hilbert scheme of points on D, its obstruction bundle is the twisted tangent bundle T_D^[n](N_D/X), and its generating series is ∏_m≥ 1(1−q^m)^−(c_2(X)· D+D^3). We work out the examples of the exceptional divisor in Bl_pℙ^3 and of a rigid section in a Fano ℙ^1-bundle over ℙ^1×ℙ^1.
ER -