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Our main result provides a best possible resolution of the monoidal structure $M_X$ of a log variety $X$ over $\\calO$ with a vertical log structure: there exists a log modification $Y\\to X$ such that the monoidal structure of $Y$ is polystable. In particular, if $X$ is log smooth over $\\mathcal{O}$, then $Y$ is polystable with a smooth generic fiber. As a corollary we deduce that any variety over $\\mathcal{O}$ possesses a polystable alteration of degreee"},"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":"1806.09168","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-06-24T15:46:13Z","cross_cats_sorted":[],"title_canon_sha256":"65f65b11e752bcebdec67d42a1aab48804ee36b9ba2d5c9e9f1f708148f06239","abstract_canon_sha256":"8361f66170b356cf1648009c3e6e717e67df6afc5c2c60426229a851f89319ad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:30.868416Z","signature_b64":"+YHlOocb+wyrDyy2jMCttvfJV3kYw0XR/LhujxSOINZEqr8h7qwJLLjrdEncNvG1dsJ7WxXKrI48pybXUZfLBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bb03c05acc9eb81f9e1a83454426025b057ff6e8078e21e42229855731032f0","last_reissued_at":"2026-05-17T23:47:30.867911Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:30.867911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Log smoothness and polystability over valuation rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Gaku Liu, Igor Pak, Karim Adiprasito, Michael Temkin","submitted_at":"2018-06-24T15:46:13Z","abstract_excerpt":"Let $\\mathcal{O}$ be a valuation ring of height one of residual characteristic exponent $p$ and with algebraically closed field of fractions. 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