{"id":"https://openalex.org/W7203775915","doi":"https://doi.org/10.48550/arxiv.2608.18054","title":"Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement","display_name":"Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement","publication_year":2026,"publication_date":"2026-08-18","ids":{"openalex":"https://openalex.org/W7203775915","doi":"https://doi.org/10.48550/arxiv.2608.18054"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.18054","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.18054","pdf_url":"https://arxiv.org/pdf/2608.18054","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.18054","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020938069","display_name":"Avik Chakravarty","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chakravarty, Avik","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044794000","display_name":"Daebeom Choi","orcid":"https://orcid.org/0009-0006-8406-8177","observed_orcids":["https://orcid.org/0009-0006-8406-8177"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Choi, Daebeom","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5147652011","display_name":"Shengjing Xu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xu, Shengjing","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9833999872207642,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9833999872207642,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.005100000184029341,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12351","display_name":"Geometry and complex manifolds","score":0.0038999998942017555,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/flops","display_name":"flops","score":0.5270000100135803},{"id":"https://openalex.org/keywords/reflexive-polytopes","display_name":"reflexive polytopes","score":0.8740000128746033},{"id":"https://openalex.org/keywords/unimodular-equivalence","display_name":"unimodular equivalence","score":0.7770000100135803},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.5820000171661377}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.835099995136261},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7229999899864197},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.6779000163078308},{"id":"https://openalex.org/C39757861","wikidata":"https://www.wikidata.org/wiki/Q1395814","display_name":"Fano plane","level":2,"score":0.6525999903678894},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.535099983215332},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5249000191688538},{"id":"https://openalex.org/C201958917","wikidata":"https://www.wikidata.org/wiki/Q1367055","display_name":"Unimodular matrix","level":2,"score":0.4964999854564667},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.4555000066757202},{"id":"https://openalex.org/C171036898","wikidata":"https://www.wikidata.org/wiki/Q256355","display_name":"Equivariant map","level":2,"score":0.44359999895095825},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4018999934196472},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.39910000562667847},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.37869998812675476},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.328000009059906},{"id":"https://openalex.org/C177846678","wikidata":"https://www.wikidata.org/wiki/Q1501864","display_name":"Projective test","level":2,"score":0.3190999925136566},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.28769999742507935},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.2759000062942505},{"id":"https://openalex.org/C5052557","wikidata":"https://www.wikidata.org/wiki/Q1528306","display_name":"Toric variety","level":2,"score":0.27399998903274536}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.18054","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.18054","pdf_url":"https://arxiv.org/pdf/2608.18054","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.18054","doi":"https://doi.org/10.48550/arxiv.2608.18054","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.18054","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.18054","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.18054","pdf_url":"https://arxiv.org/pdf/2608.18054","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7203775915.pdf","grobid_xml":"https://content.openalex.org/works/W7203775915.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4284678611"],"related_works":["https://openalex.org/W2055622362","https://openalex.org/W3106888927","https://openalex.org/W2763605391","https://openalex.org/W3103031028","https://openalex.org/W3147630765","https://openalex.org/W2798585795","https://openalex.org/W1616836920","https://openalex.org/W1669332885","https://openalex.org/W7131844754","https://openalex.org/W2963917820"],"abstract_inverted_index":{"We":[0,78,106],"disprove":[1],"Sato's":[2,104],"weak":[3,10,82,95],"\\(F\\)-equivalence":[4],"conjecture":[5,110,135],"for":[6,119,148,155],"nonsingular":[7,43],"projective":[8,23,86,92],"toric":[9,87,94],"Fano":[11,96],"varieties":[12],"in":[13,111,126],"every":[14,51,68,127],"dimension":[15],"\\(d":[16,113,149,156],"\\geq":[17,157],"3\\).":[18],"Our":[19],"counterexamples":[20,123],"are":[21],"smooth":[22],"crepant":[24],"models":[25],"of":[26,37,55,103,122,139,166],"centered":[27],"reflexive":[28,140],"simplices.":[29],"The":[30,159],"key":[31],"input":[32],"is":[33,42,48,57,146],"a":[34,58,100],"rigidity":[35],"property":[36],"ray":[38,60],"polytopes:":[39],"if":[40],"\\(X_\u03a3\\)":[41],"and":[44,46,72,98,152],"complete":[45],"\\(-K_{X_\u03a3}\\)":[47],"nef,":[49],"then":[50,79],"nonzero":[52],"lattice":[53],"point":[54],"\\(P_\u03a3=\\operatorname{Conv}(G(\u03a3))\\)":[56],"primitive":[59],"generator.":[61],"For":[62],"our":[63],"models,":[64],"this":[65,108],"rules":[66],"out":[67],"weak-Fano-preserving":[69],"equivariant":[70],"blow-up":[71],"blow-down":[73],"throughout":[74],"the":[75,120,133,137,164],"flop":[76],"class.":[77],"introduce":[80],"Gorenstein":[81,93],"\\(F\\)-equivalence,":[83],"generated":[84],"by":[85],"birational":[88],"zigzags":[89],"through":[90],"normal":[91],"varieties,":[97],"formulate":[99],"corresponding":[101],"refinement":[102],"conjecture.":[105],"prove":[107],"refined":[109,134],"dimensions":[112],"\\leq":[114,150],"3\\),":[115],"as":[116,118],"well":[117],"family":[121],"constructed":[124],"above":[125],"dimension.":[128],"Finally,":[129],"we":[130],"show":[131],"that":[132],"implies":[136],"inclusion-connectivity":[138],"\\(d\\)-polytopes":[141],"modulo":[142],"unimodular":[143],"equivalence,":[144],"which":[145],"known":[147],"4\\)":[151],"remains":[153],"open":[154],"5\\).":[158],"results":[160],"were":[161],"developed":[162],"with":[163],"assistance":[165],"GPT-5.6":[167],"Sol.":[168]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-20T00:00:00"}