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Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209, arXiv:1109.5213) define the \"derived critical locus\" Crit$(f)$, an example of a new class of spaces in derived algebraic geometry, which they call \"$-1$-shifted symplectic derived schemes\".\n  They show that intersections of algebraic Lagrangians in a smooth symplectic $\\mathbb K$-scheme, and stable moduli schemes of coherent sheaves on a Calabi-Yau 3-fold over $\\mathbb K$, are also $-1$-shifted symplectic derived schemes. Thus, their theory may hav"},"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":"1304.4508","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-04-16T16:15:22Z","cross_cats_sorted":[],"title_canon_sha256":"09804a5d48fb04f5062a21a795d00f043bc9daa809861788bf326aa1cfb7f367","abstract_canon_sha256":"98c634361f68030c74ba24b986e67b970219b78a6e91177983e23f46c32dcf50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:55.681619Z","signature_b64":"ngHaG2cgV2OWtvBxRBx8Xk4UGeTOELW4V3KFQdPVEak0DeaoeBc5ChrTtQNfvXWIkuBTGhJ9MAi7ivqAGzHMCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"231e9dfacf66930bf9f3a5324a65de1da8223353d68ffad8f296e31aeb04706f","last_reissued_at":"2026-05-18T01:30:55.680999Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:55.680999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A classical model for derived critical loci","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Dominic Joyce","submitted_at":"2013-04-16T16:15:22Z","abstract_excerpt":"Let $f:U\\to{\\mathbb A}^1$ be a regular function on a smooth scheme $U$ over a field $\\mathbb K$. 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