{"paper":{"title":"The Birational Invariance Of Fundamental Group Schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Various fundamental group schemes are birationally invariant for smooth projective varieties over perfect fields.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Hao Wang, Lingguang Li","submitted_at":"2026-04-27T03:23:31Z","abstract_excerpt":"Let $k$ be a field, $f \\colon X \\to Y$ a birational morphism of integral connected schemes proper over $k$ with $Y$ normal, $x \\in X(k)$ lying over $y \\in Y(k)$. For Tannakian categories $\\mathcal{C}_X \\subset \\mathfrak{Vect}(X)$ and $\\mathcal{C}_Y \\subset \\mathfrak{Vect}(Y)$, denote by $\\pi(\\mathcal{C}_X,x)$ and $\\pi(\\mathcal{C}_Y,y)$ the corresponding Tannaka group schemes. We establish a unified Tannakian criteria for the natural homomorphism $\\pi(\\mathcal{C}_X,x)\\to \\pi(\\mathcal{C}_Y,y)$ to be an isomorphism. As applications, for a birational map $X \\dashrightarrow Y$ between smooth projec"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"For a birational map X ⇢ Y between smooth projective varieties over a perfect field k, there exists a natural isomorphism π^*(X,x) ≅ π^*(Y,y) for any * ∈ {S,N,EN,F,EF,Loc,ELoc,ét, Eét,uni}. In particular, the induced homomorphism π^str(X,x) → π^str(Y,y) is an isomorphism for any birational morphism X → Y.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Y is normal, the schemes are integral connected and proper over k, and the Tannakian categories C_X and C_Y satisfy the general criteria making the natural homomorphism an isomorphism; for the main application the varieties must be smooth projective over a perfect field.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Various fundamental group schemes are birationally invariant for smooth projective varieties over perfect fields.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"cc27bd6393f5d574145f0d43208523624a0dedac86c0451b328e9b5d9f7b4759"},"source":{"id":"2604.23997","kind":"arxiv","version":2},"verdict":{"id":"4c48b49a-54c8-48e3-9bb8-d3d98d17274a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T02:26:25.131862Z","strongest_claim":"For a birational map X ⇢ Y between smooth projective varieties over a perfect field k, there exists a natural isomorphism π^*(X,x) ≅ π^*(Y,y) for any * ∈ {S,N,EN,F,EF,Loc,ELoc,ét, Eét,uni}. In particular, the induced homomorphism π^str(X,x) → π^str(Y,y) is an isomorphism for any birational morphism X → Y.","one_line_summary":"Various fundamental group schemes are birationally invariant for smooth projective varieties over perfect fields.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Y is normal, the schemes are integral connected and proper over k, and the Tannakian categories C_X and C_Y satisfy the general criteria making the natural homomorphism an isomorphism; for the main application the varieties must be smooth projective over a perfect field.","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.23997/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T07:40:31.961142Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:32:46.163042Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"25d22449ac312a8c90fbf068af6a8ee9e7104bfee4326239e3123cef2f10b2bf"},"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"}