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Examples of toric Fano manifolds with $rk(Pic(X)) \\leq 3$ which admit full strongly exceptional collections of line bundles were recently found by various authors. For these examples we construct a map $E : Crit(X) \\rightarrow Pic(X)$ whose image $\\mathcal{E}=\\left \\{ E(z) \\vert z \\in Crit(X) \\right \\}$ is a full strongly exceptional collection satisfying the M-aligned property. That is, under this map, the groups $Hom(E(z),E(w))$ for $z,w \\in C"},"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":"1605.00302","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-05-01T20:17:29Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"95508511805898385545d716d42715e5d125853c6efbd2d53af2bc28c44c8b8b","abstract_canon_sha256":"90fde4077e07e595bc0e4a715a59c82e50de10ca389551190639a30054233d7b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:13.825475Z","signature_b64":"hVSL8BAVwUjTyzHNaANBux9wcIS5LoZxNW28jl7+V1lvBSN1nzU2cBavR1XmTsgHfzZCJ1Y88FirmXcerS2SAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"705c21045ace2490a977ddfb9999737233bfc9645d5a32bffa0e73dedfd0966e","last_reissued_at":"2026-05-18T00:44:13.825035Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:13.825035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Landau-Ginzburg systems and $\\mathcal{D}^b(X)$ of various toric Fano manifolds with small picard group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AG","authors_text":"Yochay Jerby","submitted_at":"2016-05-01T20:17:29Z","abstract_excerpt":"For a toric Fano manifold $X$ denote by $Crit(X) \\subset (\\mathbb{C}^{\\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. 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