{
  "abstract": "In this paper we consider the general $n-2$ dimensional linear section $X$ of the Grassmannian $\\mathbb G(1,n)$ of lines in $\\mathbb{P}^n$, that is a Fano variety of index 1. We first prove that for $n$ odd $X$ is birational to a hypersurface defined by a pfaffian determinant of order $n+1$ of linear forms in $\\mathbb{P}^{n-1}$, and we study this birational transformation in some detail. Then for $n$ even, we prove that $X$ is unirational and birational to a hypersurface defined by a pfaffian determinant of order $n$ of linear forms in $\\mathbb{P}^{n-1}$. Moreover we prove that the hypersurface of $\\mathbb{P}^n$ described by the lines corresponding to the points in $X$ has degree $n-1$ and is rational, and we find a rational parametrization of it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21503",
  "author_total": 3,
  "authors": [
    {
      "name": "Michele Bolognesi"
    },
    {
      "name": "Ciro Ciliberto"
    },
    {
      "name": "Alessandro Verra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bbkedm2jpkjn4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21503",
  "fulltext_url": null,
  "id": "bbkedm2jpkjn4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21503",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213950876",
  "pdf_url": "https://arxiv.org/pdf/2609.21503",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bbkedm2jpkjn4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21503",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21503"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Fano varieties of index one that are linear sections of Grassmannians of lines",
  "title_text": "On Fano varieties of index one that are linear sections of Grassmannians of lines",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/bbkedm2jpkjn4",
  "venue": "arXiv",
  "versions": []
}