{"id":"https://openalex.org/W2340029865","doi":"https://doi.org/10.3390/sym8040027","title":"E-Polytopes in Picard Groups of Smooth Rational Surfaces","display_name":"E-Polytopes in Picard Groups of Smooth Rational Surfaces","publication_year":2016,"publication_date":"2016-04-20","ids":{"openalex":"https://openalex.org/W2340029865","doi":"https://doi.org/10.3390/sym8040027","mag":"2340029865"},"language":"en","primary_location":{"id":"doi:10.3390/sym8040027","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym8040027","pdf_url":"https://www.mdpi.com/2073-8994/8/4/27/pdf?version=1461164180","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/8/4/27/pdf?version=1461164180","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016953124","display_name":"Jae-Hyouk Lee","orcid":null},"institutions":[{"id":"https://openalex.org/I138925566","display_name":"Ewha Womans University","ror":"https://ror.org/053fp5c05","country_code":"KR","type":"education","lineage":["https://openalex.org/I138925566"]}],"countries":["KR"],"is_corresponding":true,"raw_author_name":"Jae-Hyouk Lee","raw_affiliation_strings":["Department of Mathematics, Ewha Womans University, Seoul 03760, Korea"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Ewha Womans University, Seoul 03760, Korea","institution_ids":["https://openalex.org/I138925566"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5110358979","display_name":"YongJoo Shin","orcid":null},"institutions":[{"id":"https://openalex.org/I24943067","display_name":"Fudan University","ror":"https://ror.org/013q1eq08","country_code":"CN","type":"education","lineage":["https://openalex.org/I24943067"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"YongJoo Shin","raw_affiliation_strings":["Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 103077, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 103077, China","institution_ids":["https://openalex.org/I24943067"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5016953124"],"corresponding_institution_ids":["https://openalex.org/I138925566"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.5326,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.67134649,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"8","issue":"4","first_page":"27","last_page":"27"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9994000196456909,"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":0.9994000196456909,"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":0.9970999956130981,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/polytope","display_name":"Polytope","score":0.8964556455612183},{"id":"https://openalex.org/keywords/quartic-function","display_name":"Quartic function","score":0.734941303730011},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6837062239646912},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5091671347618103},{"id":"https://openalex.org/keywords/picard-group","display_name":"Picard group","score":0.5018932819366455},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.49140608310699463},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4785381257534027},{"id":"https://openalex.org/keywords/weyl-group","display_name":"Weyl group","score":0.4592684507369995},{"id":"https://openalex.org/keywords/action","display_name":"Action (physics)","score":0.44831401109695435},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.4448709189891815},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.28762850165367126},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.17417874932289124},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15332669019699097}],"concepts":[{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.8964556455612183},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.734941303730011},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6837062239646912},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5091671347618103},{"id":"https://openalex.org/C2777283802","wikidata":"https://www.wikidata.org/wiki/Q2042963","display_name":"Picard group","level":3,"score":0.5018932819366455},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49140608310699463},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4785381257534027},{"id":"https://openalex.org/C40233952","wikidata":"https://www.wikidata.org/wiki/Q768074","display_name":"Weyl group","level":2,"score":0.4592684507369995},{"id":"https://openalex.org/C2780791683","wikidata":"https://www.wikidata.org/wiki/Q846785","display_name":"Action (physics)","level":2,"score":0.44831401109695435},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.4448709189891815},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.28762850165367126},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.17417874932289124},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15332669019699097},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym8040027","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym8040027","pdf_url":"https://www.mdpi.com/2073-8994/8/4/27/pdf?version=1461164180","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:04ac0908d9f24fceb72f48433f26c619","is_oa":true,"landing_page_url":"https://doaj.org/article/04ac0908d9f24fceb72f48433f26c619","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 8, Iss 4, p 27 (2016)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/8/4/27/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym8040027","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry; Volume 8; Issue 4; Pages: 27","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym8040027","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym8040027","pdf_url":"https://www.mdpi.com/2073-8994/8/4/27/pdf?version=1461164180","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.5699999928474426,"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions"}],"awards":[{"id":"https://openalex.org/G1079412488","display_name":null,"funder_award_id":"2013R1A1A2012783","funder_id":"https://openalex.org/F4320322120","funder_display_name":"National Research Foundation of Korea"}],"funders":[{"id":"https://openalex.org/F4320322120","display_name":"National Research Foundation of Korea","ror":"https://ror.org/013aysd81"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W659300758","https://openalex.org/W1575165434","https://openalex.org/W1632856727","https://openalex.org/W2058711712","https://openalex.org/W2090546236","https://openalex.org/W2098913191","https://openalex.org/W2148448039","https://openalex.org/W2319086082","https://openalex.org/W2490679787","https://openalex.org/W2536439311","https://openalex.org/W2616625691","https://openalex.org/W2963664828","https://openalex.org/W3101122615","https://openalex.org/W3194975564"],"related_works":["https://openalex.org/W2000297171","https://openalex.org/W4280529077","https://openalex.org/W2777256819","https://openalex.org/W4221159894","https://openalex.org/W4301265025","https://openalex.org/W2402305399","https://openalex.org/W2041579410","https://openalex.org/W4287019834","https://openalex.org/W2733444847","https://openalex.org/W3159759628"],"abstract_inverted_index":{"In":[0],"this":[1],"article,":[2],"we":[3],"introduce":[4],"special":[5],"divisors":[6],"(root,":[7],"line,":[8],"ruling,":[9],"exceptional":[10,40],"system":[11],"and":[12,19,39,51],"rational":[13,17],"quartic)":[14],"in":[15,25],"smooth":[16],"surfaces":[18],"study":[20],"their":[21],"correspondences":[22],"to":[23,44],"subpolytopes":[24],"Gosset":[26],"polytopes":[27],"k":[28,49,53],"21":[29],".":[30],"We":[31],"also":[32],"show":[33],"that":[34],"the":[35,45],"sets":[36],"of":[37,47],"rulings":[38],"systems":[41],"correspond":[42],"equivariantly":[43],"vertices":[46],"2":[48,54],"1":[50,52],"via":[55],"E-type":[56],"Weyl":[57],"action.":[58]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2016-06-24T00:00:00"}