{
  "abstract": "Let $G$ be a split Chevalley group of type $B_n$ ($n \\geq 3$) or $D_n$ ($n \\geq 4$), let $L$ be its Lie algebra and $Γ$ its long-root geometry, namely $Γ$ is the line-grassmannian of the polar space associated to $G$. It is well known that $L$ hosts a projective embedding of $Γ$, called the adjoint embedding, which realizes $Γ$ as a projective variety in the projective geometry of the underlying vector space of $L$. It is also known that $Γ$ admits the absolutely universal projective embedding, which is such that all projective embeddings of $Γ$ can be obtained from it as projections. When the field $F$ over which $G$ is defined is finite, the universal embedding of $Γ$ is just the same as the adjoint embedding. The same is not necessarily true when $F$ is infinite. Indeed, as we shall prove in this paper, the universal projective embedding of $Γ$ lives in a direct sum $K\\oplus L$ for an $F$-vector space $K$, the dual of which is isomorphic to the derivation space $\\mathrm{Der}(F)$ of $F$, which in general is non-trivial. Accordingly, the adjoint embedding of $Γ$ is universal if and only if $\\mathrm{Der}(F)$ is trivial, namely $F$ is either perfect of positive characteristic or algebraic over its minimum subfield.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33384",
  "author_total": 1,
  "authors": [
    {
      "name": "Antonio Pasini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3r4ve0cnsqrx6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33384",
  "fulltext_url": null,
  "id": "3r4ve0cnsqrx6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33384",
  "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.GR"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7223964898",
  "pdf_url": "https://arxiv.org/pdf/2610.33384",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3r4ve0cnsqrx6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33384",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33384"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Absolutely universal projective embeddings of long-root geometries of orthogonal type",
  "title_text": "Absolutely universal projective embeddings of long-root geometries of orthogonal type",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/3r4ve0cnsqrx6",
  "venue": "arXiv",
  "versions": []
}