{"paper":{"title":"On the quasi-monomiality of the $\\alpha$- and $\\delta$-invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Quasi-monomial valuations compute the alpha and delta invariants for any projective klt pair with ample line bundle, over countable or uncountable fields.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Dae-Won Lee, Donghyeon Kim","submitted_at":"2026-04-20T16:19:16Z","abstract_excerpt":"In this paper, we show that for any projective klt pair $(X,\\Delta)$ over an algebraically closed field of characteristic \\(0\\) and any big $\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisor $L$ on $X$, the invariants $\\alpha(X,\\Delta,L)$ and $\\delta(X,\\Delta,L)$ are computed by quasi-monomial valuations, without any uncountability assumption on the base field."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any projective klt pair (X,Δ) and ample line bundle L, there exist quasi-monomial valuations computing α(X,Δ,L) and δ(X,Δ,L), independently of whether the base field is countable.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The pair (X,Δ) is klt and projective with L ample; the argument likely depends on the existence of suitable models or resolutions that may require additional technical conditions from birational geometry.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Quasi-monomial valuations exist that compute α(X,Δ,L) and δ(X,Δ,L) for projective klt pairs over arbitrary fields.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Quasi-monomial valuations compute the alpha and delta invariants for any projective klt pair with ample line bundle, over countable or uncountable fields.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9ce2bd8df1fcee6163405b4c28d3e65ec1a37e66fd46ea92e982eb38b80482c7"},"source":{"id":"2604.18465","kind":"arxiv","version":3},"verdict":{"id":"e308fa2b-62da-4416-b544-28fb87eddb47","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T03:17:41.889263Z","strongest_claim":"For any projective klt pair (X,Δ) and ample line bundle L, there exist quasi-monomial valuations computing α(X,Δ,L) and δ(X,Δ,L), independently of whether the base field is countable.","one_line_summary":"Quasi-monomial valuations exist that compute α(X,Δ,L) and δ(X,Δ,L) for projective klt pairs over arbitrary fields.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The pair (X,Δ) is klt and projective with L ample; the argument likely depends on the existence of suitable models or resolutions that may require additional technical conditions from birational geometry.","pith_extraction_headline":"Quasi-monomial valuations compute the alpha and delta invariants for any projective klt pair with ample line bundle, over countable or uncountable fields."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.18465/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"}