{
  "abstract": "Let $D_{m,n}\\subset\\C^{m\\times n}$ be the union of the zero loci of the maximal minors of a generic $m\\times n$ matrix, and let $U_{m,n}$ be its complement. We show that $U_{m,n}\\cong\\GL_m(\\C)\\times\\Gr(m,n)^\\circ$, where $\\Gr(m,n)^\\circ$ is the open stratum of the Grassmannian on which all Plücker coordinates are nonzero. For $m\\in\\{2,n-2\\}$ this gives closed formulas for the topology of $U_{m,n}$ and recovers results of Yim. Our main result, which is computer-assisted, is that $D_{3,5}$ is \\emph{not} a free divisor. This answers a question of Yim. We determine $\\Der(-\\log D_{3,5})$ explicitly. It has projective dimension one, and it fails to be free exactly on the matrices of rank at most two. There the failure comes from a non-free transversal arrangement: the ten planes spanned by pairs of five general vectors in $\\C^3$. Computations in further cases support the conjecture that $D_{m,n}$ is free only for $m\\le 2$ or $n=m+1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13291",
  "author_total": 1,
  "authors": [
    {
      "name": "Marek Janasz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86d629xmpyk5f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13291",
  "fulltext_url": null,
  "id": "86d629xmpyk5f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13291",
  "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": "W7221538184",
  "pdf_url": "https://arxiv.org/pdf/2610.13291",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86d629xmpyk5f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13291",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13291"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Geometry of some determinantal arrangements",
  "title_text": "Geometry of some determinantal arrangements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/86d629xmpyk5f",
  "venue": "arXiv",
  "versions": []
}