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As a corollary, we show that if there exists a finite surjective morphism $f\\colon Y\\to X$ onto a smooth projective complex variety $X$ of Picard rank $1$ such that $(Y, f^{-1}(D)_{\\mathrm{red}})$ is a toric pair, then $X$ is the projective space and $D$ is a toric divisor."},"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":"2507.12198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2025-07-16T12:54:21Z","cross_cats_sorted":[],"title_canon_sha256":"7d3aa160cbb197112edfaa2421b73b3356ad48cff8d1260b609901f5c9268ed5","abstract_canon_sha256":"4aa49c7a126d0dd4c5601e8383aac1991d9f675305144ee95f27b9d5a940cadb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:38:16.359761Z","signature_b64":"2oyq33jWDFwWQfAdtQ2a4r6zWdTi0erMG5DiQDw337a1+i8KYyrOoJQ3SoErq37irQa7Mul8+9aMqH8cX3KOCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"132a1eb30a2396bde4143f6233c510f9169de3ec20937cf5b7f84025f08fc95b","last_reissued_at":"2026-07-05T11:38:16.359263Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:38:16.359263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Frobenius liftable hypersurfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jakub Witaszek, Supravat Sarkar, Tatsuro Kawakami","submitted_at":"2025-07-16T12:54:21Z","abstract_excerpt":"Let $D$ be a reduced divisor in $\\mathbb P^n_k$ for an algebraically closed field $k$ of positive characteristic $p > 0$. We prove that if $(\\mathbb P^n_k, D)$ is Frobenius liftable modulo $p^2$, then $D$ is a toric divisor. As a corollary, we show that if there exists a finite surjective morphism $f\\colon Y\\to X$ onto a smooth projective complex variety $X$ of Picard rank $1$ such that $(Y, f^{-1}(D)_{\\mathrm{red}})$ is a toric pair, then $X$ is the projective space and $D$ is a toric divisor."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12198","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.12198/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2507.12198","created_at":"2026-07-05T11:38:16.359323+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.12198v1","created_at":"2026-07-05T11:38:16.359323+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.12198","created_at":"2026-07-05T11:38:16.359323+00:00"},{"alias_kind":"pith_short_12","alias_value":"CMVB5MYKEOLL","created_at":"2026-07-05T11:38:16.359323+00:00"},{"alias_kind":"pith_short_16","alias_value":"CMVB5MYKEOLL3ZAU","created_at":"2026-07-05T11:38:16.359323+00:00"},{"alias_kind":"pith_short_8","alias_value":"CMVB5MYK","created_at":"2026-07-05T11:38:16.359323+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.16325","citing_title":"Images of toric variety and amplified endomorphism of weak Fano threefolds","ref_index":14,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CMVB5MYKEOLL3ZAUH5RDHRIQ7E","json":"https://pith.science/pith/CMVB5MYKEOLL3ZAUH5RDHRIQ7E.json","graph_json":"https://pith.science/api/pith-number/CMVB5MYKEOLL3ZAUH5RDHRIQ7E/graph.json","events_json":"https://pith.science/api/pith-number/CMVB5MYKEOLL3ZAUH5RDHRIQ7E/events.json","paper":"https://pith.science/paper/CMVB5MYK"},"agent_actions":{"view_html":"https://pith.science/pith/CMVB5MYKEOLL3ZAUH5RDHRIQ7E","download_json":"https://pith.science/pith/CMVB5MYKEOLL3ZAUH5RDHRIQ7E.json","view_paper":"https://pith.science/paper/CMVB5MYK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.12198&json=true","fetch_graph":"https://pith.science/api/pith-number/CMVB5MYKEOLL3ZAUH5RDHRIQ7E/graph.json","fetch_events":"https://pith.science/api/pith-number/CMVB5MYKEOLL3ZAUH5RDHRIQ7E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CMVB5MYKEOLL3ZAUH5RDHRIQ7E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CMVB5MYKEOLL3ZAUH5RDHRIQ7E/action/storage_attestation","attest_author":"https://pith.science/pith/CMVB5MYKEOLL3ZAUH5RDHRIQ7E/action/author_attestation","sign_citation":"https://pith.science/pith/CMVB5MYKEOLL3ZAUH5RDHRIQ7E/action/citation_signature","submit_replication":"https://pith.science/pith/CMVB5MYKEOLL3ZAUH5RDHRIQ7E/action/replication_record"}},"created_at":"2026-07-05T11:38:16.359323+00:00","updated_at":"2026-07-05T11:38:16.359323+00:00"}