{
  "abstract": "We leverage Polishchuk's Poisson blow-up criterion in the context of algebro-geometric integrable systems. In more detail, one may associate an integrable system $τ:\\mathfrak{X}\\longrightarrow\\mathfrak{B}$ to each affine Poisson scheme $\\mathfrak{X}$ over $\\mathbb{C}$. We prove that the blow-ups of $\\mathfrak{X}$ along fibers of $τ$ are Poisson schemes occurring in a family $\\widetilde{\\mathfrak{X}\\times\\mathfrak{B}}\\longrightarrow\\mathfrak{B}$, where $\\widetilde{\\mathfrak{X}\\times\\mathfrak{B}}$ is itself a Poisson scheme. This result is subsequently specialized to the adjoint quotient $τ:\\mathfrak{g}\\longrightarrow\\mathfrak{g}/\\!/G=:\\mathfrak{c}$ of a finite-dimensional complex semisimple Lie algebra $\\mathfrak{g}$ with integrating algebraic group $G$. We show that the family $\\widetilde{\\mathfrak{g}\\times\\mathfrak{c}}\\longrightarrow\\mathfrak{c}$ is flat, conical, and equipped with a canonical Poisson Hamiltonian $G$-variety structure. We also obtain Poisson-geometric results on the fibers of this family, which are blow-ups of $\\mathfrak{g}$ along regular adjoint orbit closures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30185",
  "author_total": 2,
  "authors": [
    {
      "name": "Peter Crooks"
    },
    {
      "name": "Iva Halacheva"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/66gp1e1e26shh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30185",
  "fulltext_url": null,
  "id": "66gp1e1e26shh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30185",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204951717",
  "pdf_url": "https://arxiv.org/pdf/2608.30185",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/66gp1e1e26shh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30185",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30185"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Poisson blow-ups and the adjoint quotient",
  "title_text": "Poisson blow-ups and the adjoint quotient",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/66gp1e1e26shh",
  "venue": "arXiv",
  "versions": []
}