{"paper":{"title":"Semistable reductions and minimalities of invariants for group scheme actions on projective schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For every point on a projective scheme under a reductive group action, the minimal invariant locus coincides with the semistable reduction translation locus.","cross_cats":["math.DS"],"primary_cat":"math.AG","authors_text":"Rin Gotou, Y\\^usuke Okuyama","submitted_at":"2026-04-28T13:57:06Z","abstract_excerpt":"Let $K$ be an algebraically closed and complete non-archimedean and non-trivially valued field, and let $G$ be a reductive group scheme acting on a flat projective scheme $X$ defined over the base ring of $K$-integers. For every $K$-point $x$ in $X$, we introduce the minimal invariant locus $\\operatorname{MinInvLoc}_x$ and the semistable reduction translation locus $\\operatorname{SSRL}_x$ in the translation space $\\operatorname{BT}_G(K)$ associated with $G_K$, which is a variant of Bruhat-Tits building, and establish not only the coincidence of those loci but, under a mild completeness assumpt"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every K-point x in X, the minimal invariant locus MinInvLoc_x and the semistable reduction translation locus SSRL_x coincide, and under a mild completeness assumption, they are non-empty.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The mild completeness assumption on the field K (in addition to being algebraically closed, complete, non-archimedean and non-trivially valued) that is required for the non-emptiness of the loci.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For reductive group scheme actions on projective schemes over complete non-archimedean fields, the minimal invariant locus coincides with the semistable reduction translation locus and is non-empty under mild completeness assumptions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For every point on a projective scheme under a reductive group action, the minimal invariant locus coincides with the semistable reduction translation locus.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"64a86aa4441fe6af0a17f133ea26271a8b1411ed3b7514d83a5eb44dcc1c5b77"},"source":{"id":"2604.25659","kind":"arxiv","version":2},"verdict":{"id":"e07b5da1-4507-442d-a740-e7da725de33a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:56:36.555784Z","strongest_claim":"For every K-point x in X, the minimal invariant locus MinInvLoc_x and the semistable reduction translation locus SSRL_x coincide, and under a mild completeness assumption, they are non-empty.","one_line_summary":"For reductive group scheme actions on projective schemes over complete non-archimedean fields, the minimal invariant locus coincides with the semistable reduction translation locus and is non-empty under mild completeness assumptions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The mild completeness assumption on the field K (in addition to being algebraically closed, complete, non-archimedean and non-trivially valued) that is required for the non-emptiness of the loci.","pith_extraction_headline":"For every point on a projective scheme under a reductive group action, the minimal invariant locus coincides with the semistable reduction translation locus."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25659/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T20:52:56.464709Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"79272e9ab10161a050159550274e6e4fa1f2672d603477847f21f1fb73894be0"},"references":{"count":18,"sample":[{"doi":"","year":2014,"title":"22, De Gruyter, Berlin, 2014","work_id":"ee96cb10-7cb2-4cab-ac24-b21253c1aad9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"296, Cambridge University Press, Cambridge, 2003","work_id":"93759958-1177-4a99-a7a3-3dc305cdb0ab","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"35, Cambridge University Press, 1997","work_id":"5a02fa77-bad3-4079-869d-818b48921e53","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2001,"title":"I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and II [ibid.; MR1295240 (95m:90002)]","work_id":"3aaeb420-f412-459a-8d82-b38bd7b5c45f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Alon Levy, The semistable reduction problem for the space of morphisms on P ^n , Algebra Number Theory 6(2012), no. 7, 1483--1501. 3007156","work_id":"559e227b-75d7-4ae2-ab83-fd00eb73a798","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":18,"snapshot_sha256":"3edb1d8c2385939c3da9317e811abf0c9917a2eeae089a90fd1c4e4c3aa76dd6","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"49c407d7b47b8b156dc114836372c2f5b447ac5467f482e28fc732a1a4120116"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}