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{
    "pk": 64947,
    "title": "Embeddings and hyperplanes of the Lie geometry \\(A_{n,\\{1,n\\}}(\\mathbb{F})\\)",
    "subtitle": null,
    "abstract": "In this paper we consider a family of projective embeddings of the geometry \\(A_{n,\\{1,n\\}}(\\mathbb{F})\\) of point-hyperplanes flags of \\(\\mathrm{PG}(n,\\mathbb{F})\\). The natural embedding \\(\\varepsilon_{\\mathrm{nat}}\\) is one of them. It maps every point-hyperplane flag \\((p,H)\\) onto the vector-line \\(\\langle {\\bf x}\\otimes\\xi\\rangle\\), where \\({\\bf x}\\) is a representative vector of \\(p\\) and \\(\\xi\\) is a linear functional describing \\(H\\). The other embeddings have been discovered more than two decads ago by Thas and Van Maldeghem for the case \\(n = 2\\) and recently generalized to any \\(n\\) by De Schepper, Schillewaert and Van Maldeghem. They are obtained as twistings of \\(\\varepsilon_{\\mathrm{nat}}\\) by non-trivial automorphisms of \\(\\mathbb{F}\\). Explicitly, for \\(\\sigma\\in \\mathrm{Aut}(\\mathbb{F})\\setminus\\{\\mathrm{id}_\\mathbb{F}\\}\\), the twisting \\(\\varepsilon_\\sigma\\) of \\(\\varepsilon_{\\mathrm{nat}}\\) by \\(\\sigma\\) maps \\((p,H)\\) onto \\(\\langle {\\bf x}^\\sigma\\otimes \\xi\\rangle\\). We shall prove that, when \\(\\mathrm{Aut}(\\mathbb{F}) \\neq \\{\\mathrm{id}_\\mathbb{F}\\}\\), a geometric hyperplane \\({\\cal H}\\) of \\(A_{n,\\{1,n\\}}(\\mathbb{F})\\) arises from \\(\\varepsilon_{\\mathrm{nat}}\\) and at least one of its twistings or from at least two distinct twistings of \\(\\varepsilon_{\\mathrm{nat}}\\) if and only if \\({\\cal H} = \\{(p,H)\\in A_{n,\\{1,n\\}}(\\mathbb{F}) \\mid p\\in A or a \\in H\\}\\) for a possibly non-incident point-hyperplane pair \\((a,A)\\) of \\(\\mathrm{PG}(n,\\mathbb{F})\\). We call these hyperplanes quasi-singular hyperplanes. With the help of this result we shall prove that if \\(|\\mathrm{Aut}(\\mathbb{F})| › 1\\) then \\(A_{n,\\{1,n\\}}(\\mathbb{F})\\) admits no absolutely universal embedding.\n \nMathematics Subject Classifications: 51A45, 20F40, 15A69\n \nKeywords: Lie geometries, Segre varieties, embeddings, hyperplanes",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Lie geometries"
        },
        {
            "word": "Segre varieties"
        },
        {
            "word": "embeddings"
        },
        {
            "word": "hyperplanes"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/0775w6m9",
    "frozenauthors": [
        {
            "first_name": "Antonio",
            "middle_name": "",
            "last_name": "Pasini",
            "name_suffix": "",
            "institution": "Department of Informatic Egineering and Mathematics, University of Siena, Siena, Italy",
            "department": ""
        }
    ],
    "date_submitted": "2024-09-26T15:46:37+02:00",
    "date_accepted": "2024-09-26T15:46:37+02:00",
    "date_published": "2024-09-30T09:00:00+02:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64947/galley/49757/download/"
        }
    ]
}