{
  "abstract": "The locus $S_d(X)$ of $d$-secant lines to a projective variety $X$ plays an important role in the study of projective varieties via projections (Lazarsfeld 1987, Kwak 1998, Beheshti-Eisenbud 2010). For a projective scheme $X\\subseteq\\mathbb{P}^r$ satisfying $\\textbf{N}_{d,2}$, we show that $S_d(X)\\cup X$ is cut out set-theoretically by the $r$-minors of a matrix $M$ constructed from $d$-forms scheme-theoretically defining $X$ and their linear syzygies. In particular, $S_d(X)=\\mathbb{P}^r$ if and only if every $r$-minor of $M$ vanishes. Using this, we find a singular threefold $X\\subseteq\\mathbb{P}^5$ with $S_4(X)\\ne \\mathbb{P}^5$ and $(I_X)_3=0$, and a normal fourfold $X\\subseteq\\mathbb{P}^7$ with $S_3(X)\\ne \\mathbb{P}^7$ and $(I_X)_2=0$. The nonexistence of such varieties in the smooth case is an open question raised by the second author and by Gruson-Peskine. Next, we consider the locus $S_{k,d}(X)$ of $k$-planes $L$ such that $L\\cap X$ contains a hypersurface of degree $\\ge d$ in $L$ so that $S_{1,d}(X)=S_d(X)$. The locus $S_{k,d}(X)\\cup X$ is set-theoretically defined by the $(r+1-k)$-minors of $M$. As a result, we obtain determinantal equations vanishing on the $(q+1)$-secant variety $σ_{q+1}X$ from the equations of $σ_qX$ and their linear syzygies when $σ_qX$ satisfies $\\textbf{N}_{q+1,2}$. Finally, we extend the $d$-secant lemma for lines to the locus $S_{k,d}(X)$ of $k$-planes. This yields a lower bound on the number of linear syzygies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12078",
  "author_total": 2,
  "authors": [
    {
      "name": "Jong In Han"
    },
    {
      "name": "Sijong Kwak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cf2685fr2mkce.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12078",
  "fulltext_url": null,
  "id": "cf2685fr2mkce",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12078",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221136932",
  "pdf_url": "https://arxiv.org/pdf/2610.12078",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cf2685fr2mkce.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12078",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12078"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Linear syzygies and linear subspaces whose lines are multisecant",
  "title_text": "Linear syzygies and linear subspaces whose lines are multisecant",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/cf2685fr2mkce",
  "venue": "arXiv",
  "versions": []
}