{
  "abstract": "We present a framework for studying the moduli space of linear determinantal representations for flat families of complex projective cubic surfaces across singularity boundaries. First, we deal with the classical deformation theory where a surface S_0 possesses a rational double point (RDP) of type E_6. By establishing a global simultaneous resolution over a finite ramified Galois covering of the global parameter slice, we realize the relative moduli space H_N as a diagonal Weyl quotient (h x R)/W(E_6). Second, we extend this classification to the critical boundary configuration where S0 contains a unique isolated simple elliptic singularity of type E~_6. Because the local monodromy group becomes infinite, the classical simultaneous resolution framework completely breaks down. We bypass this obstruction by constructing a geometric substitute Z derived from the filtration of Looijenga's invariant algebra of affine Weyl groups. We also show that the vector bundle Z decomposes into a direct sum of line bundles over the structural elliptic curve E, with degrees explicitly matching the negative Coxeter marks of the highest root of E_6. Utilizing Riemann's extension theorem and vector bundle rigidity over elliptic curves, we settle our isomorphism result, proving that the moduli space H-bar representing the semiuniversal family is globally isomorphic to the total space of Z.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25695",
  "author_total": 1,
  "authors": [
    {
      "name": "Patrick Omukuba"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6vsknfwe4j87y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25695",
  "fulltext_url": null,
  "id": "6vsknfwe4j87y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25695",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204472545",
  "pdf_url": "https://arxiv.org/pdf/2608.25695",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6vsknfwe4j87y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.25695",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25695"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Moduli Space of Determinantal Representations of Cubic Surfaces and Invariant Theory of Root Systems",
  "title_text": "The Moduli Space of Determinantal Representations of Cubic Surfaces and Invariant Theory of Root Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/6vsknfwe4j87y",
  "venue": "arXiv",
  "versions": []
}