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When $X$ is minimal and its Kodaira dimension is positive, this sequence of flips terminates in $M(H_X)$; $H_X$ is an ample line bundle lying so closely to $K_X$ that the canonical divisor of $M(H_X)$ is nef. Remark that so-called Thaddeus-type flips somewhat differ from f"},"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":"0811.3522","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2008-11-21T11:45:09Z","cross_cats_sorted":[],"title_canon_sha256":"4711a0ee1720b13bc2af8b020925464db42d844cd680b393e002147d3e11fa63","abstract_canon_sha256":"215e63e6dec7d554f7970492281a11d373a57d6cf1947086bc098fa08fcca84d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:34:57.813736Z","signature_b64":"4Gqb11s/+eKFj6PGf3TVhnrMICi3eRXmcRVWlY1XhaY+Bvydei3S2dUz8KZlca7YC5UHj1QbjmAjNeT0/Rp2AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b8e53ef60882b0d2ba936b4dff8fbff28c7a82b3919807aab2b75e85c4cf074","last_reissued_at":"2026-07-04T15:34:57.813298Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:34:57.813298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Flips and variation of moduli schemes of sheaves on a surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Kimiko Yamada","submitted_at":"2008-11-21T11:45:09Z","abstract_excerpt":"Let $H$ be an ample line bundle on a non-singular projective surface $X$, and $M(H)$ the coarse moduli scheme of rank-two $H$-semistable sheaves with fixed Chern classes on $X$. We show that if $H$ changes and passes through walls to get closer to $K_X$, then $M(H)$ undergoes natural flips with respect to canonical divisors. When $X$ is minimal and its Kodaira dimension is positive, this sequence of flips terminates in $M(H_X)$; $H_X$ is an ample line bundle lying so closely to $K_X$ that the canonical divisor of $M(H_X)$ is nef. 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