{
  "abstract": "We develop a general first-order theory of Schur apolarity for the study of secant varieties of flag varieties in arbitrary homogeneous embeddings. Extending the classical apolarity--fat-point correspondence for Veronese varieties, we show that in the Schur setting the algebraic square of the apolar ideal need not coincide with the geometric double-point conditions. We introduce a geometric Schur square whose relevant component is the conormal space, yielding a Schur Dual Terracini Lemma. Our construction recovers classical apolarity in the symmetric case. A slot-by-slot Consistency Theorem realizes these intrinsic conditions as multigraded double points. As an application, we determine the dimensions of all secant varieties of $\\operatorname{Fl}(1,2;V_n)$ embedded by $\\mathcal{O}(1,1)$: the only defective cases are $σ_2(\\operatorname{Fl}(1,2;V_3))$ and $σ_3(\\operatorname{Fl}(1,2;V_4))$, both of defect one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09301",
  "author_total": 3,
  "authors": [
    {
      "name": "Alessandra Bernardi"
    },
    {
      "name": "Stefano Canino"
    },
    {
      "name": "Vincenzo Antonio Isoldi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6wk0y8efqwp4j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09301",
  "fulltext_url": null,
  "id": "6wk0y8efqwp4j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09301",
  "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.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212146097",
  "pdf_url": "https://arxiv.org/pdf/2609.09301",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6wk0y8efqwp4j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09301",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09301"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Secant varieties of flag varieties via Schur apolarity",
  "title_text": "Secant varieties of flag varieties via Schur apolarity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/6wk0y8efqwp4j",
  "venue": "arXiv",
  "versions": []
}