{
  "abstract": "Let $X=\\mathbb{P}^n$, where $p=n+1$ is prime. Consider the orbit of the standard numerical exceptional basis in $K_0(X)$ under mutations, sign changes, and isometries of the Euler form. We construct an invariant of this orbit: for every basis from the orbit, the square of rank of all basic vectors art congruent to $1$ modulo $p$. For $\\mathbb{P}^4$ and $\\mathbb{P}^6$, we present explicit numerical exceptional bases violating this congruence; hence the action is not transitive. We also establish a stronger obstruction modulo $p^2$ and use Polishchuk's theorem to show that no vector of the constructed bases is the class of an exceptional object. Finally, we prove that for every prime $p$, the category $D^b(\\operatorname{Coh}\\mathbb{P}^{p-1}_{\\mathbb C})$ does not contain a pair of mutually orthogonal exceptional objects.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31413",
  "author_total": 1,
  "authors": [
    {
      "name": "Roman Eliseev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2gp51qkq3741k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31413",
  "fulltext_url": null,
  "id": "2gp51qkq3741k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31413",
  "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"
    ],
    [
      "arxiv",
      "math.KT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214646817",
  "pdf_url": "https://arxiv.org/pdf/2609.31413",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2gp51qkq3741k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31413",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31413"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Nontransitivity of braid group actions on exceptional bases in $K_0(\\mathbb{P}^n)$ for $n=4,6$",
  "title_text": "Nontransitivity of braid group actions on exceptional bases in K_0(ℙ^n) for n=4,6",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/2gp51qkq3741k",
  "venue": "arXiv",
  "versions": []
}