{
  "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 $α_n$ is stable and uniquely of the form $ι_{P*}\\mathcal{I}_Z$ 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 $\\operatorname{Bl}_p\\mathbb{P}^3$ and of a rigid section in a Fano $\\mathbb{P}^1$-bundle over $\\mathbb{P}^1\\times\\mathbb{P}^1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28501",
  "author_total": 1,
  "authors": [
    {
      "name": "Reginald Anderson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4bjw5d0kp5ewh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28501",
  "fulltext_url": null,
  "id": "4bjw5d0kp5ewh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28501",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.28501",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4bjw5d0kp5ewh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28501",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28501"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Planar rank-one sheaves on $\\mathbb{P}^3$, obstruction bundles, and divisor-supported Donaldson--Thomas series",
  "title_text": "Planar rank-one sheaves on ℙ^3, obstruction bundles, and divisor-supported Donaldson--Thomas series",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/4bjw5d0kp5ewh",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.13140/rg.2.2.22799.04003",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}