{
  "abstract": "Let $V$ be a complex vector space of dimension $mk$ ($m\\ge2$). A morphism from a positive-dimensional projective variety $f\\colon X\\to\\mathrm{Gr}(k,V)$ is $m$-direct-sum if $V=U_{x_1}\\oplus\\cdots\\oplus U_{x_m}$ for every $m$ pairwise distinct points $x_1,\\dots,x_m$ on $X$, where $U_x\\subset V$ is the subspace represented by $f(x)$. We show that $X$ is necessarily a smooth rational curve. When the so-called drill $d(f)$ of $f$ is $1$ or when the rank $k$ is at most $2$, we give complete classifications of $f$ up to the automorphisms of $\\mathrm{Gr}(k,V)$. As applications, we classify all so-called skew scrolls in $\\mathbb{P}^3$, which completely answers a question of Day. Then we show that if $X\\subset \\mathbb{P}(V)$ is a smooth, projective, nondegenerate subvariety whose higher-order Terracini locus is empty, then $X$ is the rational normal curve, which strengthens the critical Terracini rigidity discovered by Ballico and Chiantini. By using the same machinery, we also prove a dual version of Terracini rigidity. Finally, we show that $m$-unisolvent systems form an abelian category of finite length and that $7$ is the least diagonal exponent of a simple rank-three $2$-unisolvent system.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38184",
  "author_total": 3,
  "authors": [
    {
      "name": "Duo Li"
    },
    {
      "name": "Yanjie Li"
    },
    {
      "name": "Francesco Russo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fqxgzgt62cy8r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38184",
  "fulltext_url": null,
  "id": "fqxgzgt62cy8r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38184",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38184",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fqxgzgt62cy8r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38184",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38184"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Geometry of $m$-direct-sum morphisms",
  "title_text": "Geometry of m-direct-sum morphisms",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/fqxgzgt62cy8r",
  "venue": "arXiv",
  "versions": []
}