{
  "abstract": "Essential dimension counts the parameters needed to define a torsor, but it can drop sharply over a field extension, and it does not say how large an extension is needed. For a $G$-torsor $τ$ over a field $F\\supseteq k$ we study the \\emph{finite-degree profile} $\\edd d_k(τ)=\\min\\{\\ed_k(τ\\otimes_FF'):[F':F]\\le d\\}$, a variant of invariants of Farb--Kisin--Wolfson and Farb--Wolfson, and its jump degrees $d_j(τ)$, the least degree of an extension over which $τ$ needs only $j$ parameters. The profile is geometric over every field: $\\edd d_k(τ)\\le j$ iff some generically free $G$-variety of dimension $\\le j$ has a closed point of degree $\\le d$ in the free locus of its twist by $τ$. We prove a Sylow divisibility theorem for jump degrees and a valuation-theoretic rank--index theorem, $\\edd d_k(\\tgen)\\ge\\min\\{\\rk B:B\\le A,\\ [A:B]\\le d\\}$ for abelian $A$ of order prime to $\\operatorname{char}k$, which determines the profile of every abelian group over a field with enough roots of unity, for instance $n-\\lfloor\\log_2d\\rfloor$ for $(\\Z/2)^n$. For $\\Z/p$ in characteristic $0$ we show $\\lceil m/2\\rceil\\le d_1\\le[k(ζ_p+ζ_p^{-1}):k]$, $m=[k(ζ_p):k]$, with equality for $m$ even. For $S_5$ in characteristic $0$ we prove $d_1=4$, so Klein's reduction of the generic quintic is optimal; the lower bound comes from Abel--Jacobi rigidity, not from cohomological invariants. For $n\\ge6$, $6\\mid d_1(S_n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07508",
  "author_total": 1,
  "authors": [
    {
      "name": "Abhishek Shukla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e89e50zznn4gb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07508",
  "fulltext_url": null,
  "id": "e89e50zznn4gb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07508",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220974835",
  "pdf_url": "https://arxiv.org/pdf/2610.07508",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e89e50zznn4gb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07508",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07508"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The finite-degree profile of essential dimension I: general theory and characteristic prime to $|G|$",
  "title_text": "The finite-degree profile of essential dimension I: general theory and characteristic prime to |G|",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/e89e50zznn4gb",
  "venue": "arXiv",
  "versions": []
}