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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.2577","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-12-09T20:53:20Z","cross_cats_sorted":[],"title_canon_sha256":"ae5cda7ac1e0c6fbd976811eaf2856825abb936ee50d8ccfe287c216022b663d","abstract_canon_sha256":"898dcb38f4ff4597654ef774e3b966ae7d8ab806d2472dd3ed3ca22928bde8b7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:24.605540Z","signature_b64":"Gg9LLf4oUZkgJefYXvrGMPEXLpePDcEZy8/RBz5dmWdbKrspq0+gaK65tudevkupYk8ezR1dLsrcxLS1AAfLDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2908fe217b0c9c00c05ccebd5d6117f93e454d15863719cda6931acdeab9540","last_reissued_at":"2026-05-18T01:22:24.604881Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:24.604881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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