{"paper":{"title":"A result on the generic Picard number of surfaces in fake weighted projective 3-spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A criterion based on edge degenerations shows that certain generic nondegenerate surfaces in fake weighted projective 3-spaces have Picard number greater than 1.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Julius Giesler","submitted_at":"2026-04-28T14:34:30Z","abstract_excerpt":"We give a criterion for certain generic nondegenerate surfaces in a fake weighted projective $3$-space to have Picard number $>1$. These algebraic surfaces are of general type. We do this by considering degenerations (along an edge), keeping track of the geometric genus, and using vanishing cohomology classes to construct a rational Picard class on the surface not proportional to the canonical divisor."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give a criterion for certain generic nondegenerate surfaces in a fake weighted projective 3-space to have Picard number >1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That degenerations along an edge permit reliable tracking of the geometric genus and that vanishing cohomology classes produce a rational Picard class independent of the canonical divisor.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A criterion is established for generic nondegenerate surfaces in fake weighted projective 3-spaces to have Picard number >1 via edge degenerations, genus tracking, and vanishing cohomology classes yielding an extra rational Picard class.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A criterion based on edge degenerations shows that certain generic nondegenerate surfaces in fake weighted projective 3-spaces have Picard number greater than 1.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d0f7d526904e05861b68a2e9d84d9fd852377afaad0fe193b98477ccf5563d8b"},"source":{"id":"2604.25709","kind":"arxiv","version":2},"verdict":{"id":"009fe047-a6d7-4bc8-bbfe-166ab220e7a2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T15:06:31.398064Z","strongest_claim":"We give a criterion for certain generic nondegenerate surfaces in a fake weighted projective 3-space to have Picard number >1.","one_line_summary":"A criterion is established for generic nondegenerate surfaces in fake weighted projective 3-spaces to have Picard number >1 via edge degenerations, genus tracking, and vanishing cohomology classes yielding an extra rational Picard class.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That degenerations along an edge permit reliable tracking of the geometric genus and that vanishing cohomology classes produce a rational Picard class independent of the canonical divisor.","pith_extraction_headline":"A criterion based on edge degenerations shows that certain generic nondegenerate surfaces in fake weighted projective 3-spaces have Picard number greater than 1."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25709/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T04:35:16.807567Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:51:41.709930Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b8dfa5b7c9b6a0a4815bd9feee9d65fe9670ab96368bc09778be952289409db1"},"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"}