{
  "abstract": "We study finite determinacy and modality in positive characteristic through three complementary constructions: prescribed-jet transversality, Frobenius deformation directions, and a finite flat critical cover. For matrix germs over an arbitrary field, prescribed-jet transversality to the rank stratification shows that finite left-right determinacy is equivalent to finite codimension of the tangent image. Analogous results hold for pure right equivalence, for one-sided actions over infinite fields, and in specified ranges over finite fields. These methods also yield a characterization of finite contact determinacy for positive-dimensional ideals and imply expected-height, Cohen-Macaulayness, reducedness, and normality results for determinantal ideals. For isolated hypersurface singularities over an algebraically closed field of characteristic $p>0$, we introduce a Frobenius algebra whose length and Loewy structure govern the Milnor number, determinacy, and essential corank. Using a finite flat critical cover, we prove that proper modality coincides with right modality. It follows that, in fixed characteristic, bounded right modality gives an ambient-dimension-independent bound on determinacy and, up to nonsingular quadratic suspension, a finite algebraic parametrization by modular families. The same Frobenius methods yield uniform finiteness results for $F$-jumping numbers and show that the $F$-pure threshold is determined at a single Frobenius level.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07092",
  "author_total": 1,
  "authors": [
    {
      "name": "Yotam Svoray"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6zkhffkjqef2t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07092",
  "fulltext_url": null,
  "id": "6zkhffkjqef2t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07092",
  "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.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220895156",
  "pdf_url": "https://arxiv.org/pdf/2610.07092",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6zkhffkjqef2t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07092",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07092"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Determinacy and Modality in Positive Characteristic",
  "title_text": "Determinacy and Modality in Positive Characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/6zkhffkjqef2t",
  "venue": "arXiv",
  "versions": []
}