Pub Date : 2025-01-01Epub Date: 2025-09-18DOI: 10.1007/s00022-025-00772-7
Miroslav Korbelář, Jan Paseka, Thomas Vetterlein
An orthoset is a set equipped with a symmetric, irreflexive binary relation. With any (anisotropic) Hermitian space H, we may associate the orthoset , consisting of the set of one-dimensional subspaces of H and the usual orthogonality relation. determines H essentially uniquely.We characterise in this paper certain kinds of Hermitian spaces by imposing transitivity and minimality conditions on their associated orthosets. By gradually considering stricter conditions, we restrict the discussion to a narrower and narrower class of Hermitian spaces. Ultimately, our interest lies in quadratic spaces over countable subfields of .A line of an orthoset is the orthoclosure of two distinct elements. For an orthoset to be line-symmetric means roughly that its automorphism group acts transitively both on the collection of all lines as well as on each single line. Line-symmetric orthosets turn out to be in correspondence with transitive Hermitian spaces. Furthermore, quadratic orthosets are defined similarly, but are required to possess, for each line , a group of automorphisms acting on transitively and commutatively. We show the correspondence of quadratic orthosets with transitive quadratic spaces over ordered fields. We finally specify those quadratic orthosets that are, in a natural sense, minimal: for a finite , the orthoset , where R is the Hilbert field, has the property of being embeddable into any other quadratic orthoset of rank n.
{"title":"Characterisation of quadratic spaces over the Hilbert field by means of the orthogonality relation.","authors":"Miroslav Korbelář, Jan Paseka, Thomas Vetterlein","doi":"10.1007/s00022-025-00772-7","DOIUrl":"10.1007/s00022-025-00772-7","url":null,"abstract":"<p><p>An orthoset is a set equipped with a symmetric, irreflexive binary relation. With any (anisotropic) Hermitian space <i>H</i>, we may associate the orthoset <math><mrow><mo>(</mo> <mi>P</mi> <mo>(</mo> <mi>H</mi> <mo>)</mo> <mo>,</mo> <mo>⊥</mo> <mo>)</mo></mrow> </math> , consisting of the set of one-dimensional subspaces of <i>H</i> and the usual orthogonality relation. <math><mrow><mo>(</mo> <mi>P</mi> <mo>(</mo> <mi>H</mi> <mo>)</mo> <mo>,</mo> <mo>⊥</mo> <mo>)</mo></mrow> </math> determines <i>H</i> essentially uniquely.We characterise in this paper certain kinds of Hermitian spaces by imposing transitivity and minimality conditions on their associated orthosets. By gradually considering stricter conditions, we restrict the discussion to a narrower and narrower class of Hermitian spaces. Ultimately, our interest lies in quadratic spaces over countable subfields of <math><mi>R</mi></math> .A line of an orthoset is the orthoclosure of two distinct elements. For an orthoset to be line-symmetric means roughly that its automorphism group acts transitively both on the collection of all lines as well as on each single line. Line-symmetric orthosets turn out to be in correspondence with transitive Hermitian spaces. Furthermore, quadratic orthosets are defined similarly, but are required to possess, for each line , a group of automorphisms acting on transitively and commutatively. We show the correspondence of quadratic orthosets with transitive quadratic spaces over ordered fields. We finally specify those quadratic orthosets that are, in a natural sense, minimal: for a finite <math><mrow><mi>n</mi> <mo>⩾</mo> <mn>4</mn></mrow> </math> , the orthoset <math><mrow><mo>(</mo> <mi>P</mi> <mrow><mo>(</mo> <msup><mi>R</mi> <mi>n</mi></msup> <mo>)</mo></mrow> <mo>,</mo> <mo>⊥</mo> <mo>)</mo></mrow> </math> , where <i>R</i> is the Hilbert field, has the property of being embeddable into any other quadratic orthoset of rank <i>n</i>.</p>","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":"116 3","pages":"33"},"PeriodicalIF":0.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12446126/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145113326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1007/s00022-023-00702-5
Arne Botteldoorn, Kris Coolsaet, V. Fack
{"title":"Classification of minimal blocking sets in PG(2,9)","authors":"Arne Botteldoorn, Kris Coolsaet, V. Fack","doi":"10.1007/s00022-023-00702-5","DOIUrl":"https://doi.org/10.1007/s00022-023-00702-5","url":null,"abstract":"","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":"59 15","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138956943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-18DOI: 10.1007/s00022-023-00700-7
Ralph Howard
{"title":"Schur type comparison theorems for affine curves","authors":"Ralph Howard","doi":"10.1007/s00022-023-00700-7","DOIUrl":"https://doi.org/10.1007/s00022-023-00700-7","url":null,"abstract":"","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":" 36","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138994890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-15DOI: 10.1007/s00022-023-00697-z
Duy Ho
{"title":"On toroidal circle planes with groups of automorphisms fixing exactly one point","authors":"Duy Ho","doi":"10.1007/s00022-023-00697-z","DOIUrl":"https://doi.org/10.1007/s00022-023-00697-z","url":null,"abstract":"","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":"100 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139272283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1007/s00022-023-00695-1
Yusei Aoki, Toshiaki Adachi
{"title":"Characteristic magnetic focal values on Kähler manifolds","authors":"Yusei Aoki, Toshiaki Adachi","doi":"10.1007/s00022-023-00695-1","DOIUrl":"https://doi.org/10.1007/s00022-023-00695-1","url":null,"abstract":"","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":"75 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135221099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1007/s00022-023-00694-2
Ren Takagi
{"title":"$$phi $$-Sectional curvatures of homogeneous real hypersurfaces in a complex projective space","authors":"Ren Takagi","doi":"10.1007/s00022-023-00694-2","DOIUrl":"https://doi.org/10.1007/s00022-023-00694-2","url":null,"abstract":"","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-06DOI: 10.1007/s00022-023-00692-4
Vladimir Volenec, Ema Jurkin, Marija Šimić Horvath
{"title":"On quadruples of orthopoles","authors":"Vladimir Volenec, Ema Jurkin, Marija Šimić Horvath","doi":"10.1007/s00022-023-00692-4","DOIUrl":"https://doi.org/10.1007/s00022-023-00692-4","url":null,"abstract":"","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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