{"id":"https://openalex.org/W7204769823","doi":"https://doi.org/10.48550/arxiv.2608.27696","title":"Finding large families of rational curves through Bend-and-Break","display_name":"Finding large families of rational curves through Bend-and-Break","publication_year":2026,"publication_date":"2026-08-27","ids":{"openalex":"https://openalex.org/W7204769823","doi":"https://doi.org/10.48550/arxiv.2608.27696"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.27696","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27696","pdf_url":"https://arxiv.org/pdf/2608.27696","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/2608.27696","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006811047","display_name":"Eric Jovinelly","orcid":"https://orcid.org/0000-0001-9183-0699","observed_orcids":["https://orcid.org/0000-0001-9183-0699"]},"institutions":[{"id":"https://openalex.org/I103531236","display_name":"Boston College","ror":"https://ror.org/02n2fzt79","country_code":"US","type":"education","lineage":["https://openalex.org/I103531236"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jovinelly, Eric","raw_affiliation_strings":["Department of Mathematics , Boston College , Chestnut Hill , MA 02467"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Boston College , Chestnut Hill , MA 02467","institution_ids":["https://openalex.org/I103531236"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5123099207","display_name":"Brian Lehmann","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I103531236","display_name":"Boston College","ror":"https://ror.org/02n2fzt79","country_code":"US","type":"education","lineage":["https://openalex.org/I103531236"]},{"id":"https://openalex.org/I107639228","display_name":"University of Notre Dame","ror":"https://ror.org/00mkhxb43","country_code":"US","type":"education","lineage":["https://openalex.org/I107639228"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lehmann, Brian","raw_affiliation_strings":["Department of Mathematics , Boston College , Chestnut Hill , MA 02467","Department of Mathematics , University of Notre Dame , 255 Hurley Hall , Notre Dame , IN 46556"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Boston College , Chestnut Hill , MA 02467","institution_ids":["https://openalex.org/I103531236"]},{"raw_affiliation_string":"Department of Mathematics , University of Notre Dame , 255 Hurley Hall , Notre Dame , IN 46556","institution_ids":["https://openalex.org/I107639228"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073118156","display_name":"Eric Riedl","orcid":"https://orcid.org/0000-0003-0258-5682","observed_orcids":["https://orcid.org/0000-0003-0258-5682"]},"institutions":[{"id":"https://openalex.org/I103531236","display_name":"Boston College","ror":"https://ror.org/02n2fzt79","country_code":"US","type":"education","lineage":["https://openalex.org/I103531236"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Riedl, Eric","raw_affiliation_strings":["Department of Mathematics , Boston College , Chestnut Hill , MA 02467"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Boston College , Chestnut Hill , MA 02467","institution_ids":["https://openalex.org/I103531236"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"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":0.9994999766349792,"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":0.9994999766349792,"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/T12351","display_name":"Geometry and complex manifolds","score":0.0003000000142492354,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/rational-curves","display_name":"rational curves","score":0.9480000138282776},{"id":"https://openalex.org/keywords/fano-hypersurfaces","display_name":"Fano hypersurfaces","score":0.9599999785423279},{"id":"https://openalex.org/keywords/families-of-curves","display_name":"families of curves","score":0.609000027179718},{"id":"https://openalex.org/keywords/higher-genus-curves","display_name":"higher genus curves","score":0.8199999928474426}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8151000142097473},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.6923999786376953},{"id":"https://openalex.org/C39757861","wikidata":"https://www.wikidata.org/wiki/Q1395814","display_name":"Fano plane","level":2,"score":0.6330000162124634},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.6227999925613403},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.5755000114440918},{"id":"https://openalex.org/C22891489","wikidata":"https://www.wikidata.org/wiki/Q380770","display_name":"Fujita scale","level":2,"score":0.5687999725341797},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5020999908447266},{"id":"https://openalex.org/C157369684","wikidata":"https://www.wikidata.org/wiki/Q34740","display_name":"Genus","level":2,"score":0.4447000026702881},{"id":"https://openalex.org/C53622353","wikidata":"https://www.wikidata.org/wiki/Q1518913","display_name":"Family of curves","level":2,"score":0.4339999854564667},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.42010000348091125},{"id":"https://openalex.org/C96881808","wikidata":"https://www.wikidata.org/wiki/Q1879333","display_name":"Birational geometry","level":2,"score":0.39719998836517334},{"id":"https://openalex.org/C37500760","wikidata":"https://www.wikidata.org/wiki/Q7858555","display_name":"Twisted cubic","level":5,"score":0.35350000858306885},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32580000162124634},{"id":"https://openalex.org/C2777382242","wikidata":"https://www.wikidata.org/wiki/Q6017816","display_name":"Index (typography)","level":2,"score":0.2872999906539917},{"id":"https://openalex.org/C134949993","wikidata":"https://www.wikidata.org/wiki/Q2068617","display_name":"Pencil (optics)","level":2,"score":0.2775000035762787},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.2757999897003174},{"id":"https://openalex.org/C156330621","wikidata":"https://www.wikidata.org/wiki/Q3393476","display_name":"Rational point","level":3,"score":0.2736999988555908},{"id":"https://openalex.org/C68363185","wikidata":"https://www.wikidata.org/wiki/Q180969","display_name":"Algebraic geometry","level":2,"score":0.26919999718666077},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2685000002384186},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26159998774528503},{"id":"https://openalex.org/C75190567","wikidata":"https://www.wikidata.org/wiki/Q41237","display_name":"Rational function","level":2,"score":0.2558000087738037}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.27696","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27696","pdf_url":"https://arxiv.org/pdf/2608.27696","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.2608.27696","doi":"https://doi.org/10.48550/arxiv.2608.27696","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.27696","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:2608.27696","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27696","pdf_url":"https://arxiv.org/pdf/2608.27696","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/W7204769823.pdf","grobid_xml":"https://content.openalex.org/works/W7204769823.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4406776299","https://openalex.org/W4415914396"],"related_works":["https://openalex.org/W2470043697","https://openalex.org/W7096501765","https://openalex.org/W1905097685","https://openalex.org/W2948419592","https://openalex.org/W3088033427","https://openalex.org/W4406418343","https://openalex.org/W4402954096","https://openalex.org/W2065512620","https://openalex.org/W3046188249","https://openalex.org/W2546453482"],"abstract_inverted_index":{"We":[0,51],"present":[1],"a":[2,73],"new":[3],"construction":[4,21],"that":[5,39,71],"allows":[6],"us":[7],"to":[8],"break":[9],"off":[10],"large-degree":[11],"rational":[12,28,49,55],"curves":[13,29,56,87],"from":[14],"families":[15,47],"of":[16,32,48,66,77,84,88],"higher":[17],"genus":[18,85],"curves.":[19,50],"Our":[20],"and":[22,30],"results":[23],"deepen":[24],"the":[25,33,93],"connection":[26],"between":[27],"positivity":[31],"anticanonical":[34],"divisor.":[35],"Specifically,":[36],"we":[37,69],"show":[38],"varieties":[40],"with":[41],"large":[42,46,90],"Fujita":[43],"invariant":[44],"admit":[45],"also":[52],"construct":[53],"free":[54],"on":[57],"certain":[58],"singular":[59],"Fano":[60,75],"varieties.":[61],"As":[62],"an":[63],"explicit":[64],"consequence":[65],"our":[67],"results,":[68],"prove":[70],"for":[72],"general":[74],"hypersurface":[76],"index":[78],"at":[79],"least":[80],"3,":[81],"all":[82],"spaces":[83],"g":[86],"sufficiently":[89],"degree":[91],"have":[92],"expected":[94],"dimension.":[95]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-01T00:00:00"}