{"paper":{"title":"Fano schemes of determinants and permanents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Melody Chan, Nathan Ilten","submitted_at":"2013-12-09T20:53:20Z","abstract_excerpt":"Let $D_{m,n}^r$ and $P_{m,n}^r$ denote the subschemes of $\\mathbb{P}^{mn-1}$ given by the $r\\times r$ determinants (respectively the $r\\times r$ permanents) of an $m\\times n$ matrix of indeterminates. In this paper, we study the geometry of the Fano schemes $\\mathbf{F}_k(D_{m,n}^r)$ and $\\mathbf{F}_k(P_{m,n}^r)$ parametrizing the $k$-dimensional planes in $\\mathbb{P}^{mn-1}$ lying on $D_{m,n}^r$ and $P_{m,n}^r$, respectively. We prove results characterizing which of these Fano schemes are smooth, irreducible, and connected; and we give examples showing that they need not be reduced. We show th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2577","kind":"arxiv","version":2},"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"}