{"paper":{"title":"A geometric approach to alternating $k$-linear forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Antonio Pasini, Ilaria Cardinali, Luca Giuzzi","submitted_at":"2016-01-29T14:08:00Z","abstract_excerpt":"Given an $n$-dimensional vector space $V$ over a field $\\mathbb K$, let $2\\leq k < n$. There is a natural correspondence between the alternating $k$-linear forms $\\varphi$ of $V$ and the linear functionals $f$ of $\\bigwedge^kV$. Let $\\varepsilon_k:{\\mathcal G}_k(V)\\rightarrow {\\mathrm{PG}}(\\bigwedge^kV)$ be the Plucker embedding of the $k$-Grassmannian ${\\mathcal G}_k(V)$ of $V$. Then $\\varepsilon_k^{-1}(\\ker(f)\\cap\\varepsilon_k(\\mathcal{G}_k(V)))$ is a hyperplane of the point-line geometry ${\\mathcal G}_k(V)$. All hyperplanes of ${\\mathcal G}_k(V)$ can be obtained in this way. For a hyperplan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.08115","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}