Journal of GeometryPub Date : 2025-01-01Epub Date: 2025-09-18DOI: 10.1007/s00022-025-00772-7
Miroslav Korbelář, Jan Paseka, Thomas Vetterlein
{"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}
{"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}
{"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}
{"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}
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}
{"title":"Analytic classification of toric Kähler instanton metrics in dimension 4","authors":"Brian Weber","doi":"10.1007/s00022-023-00689-z","DOIUrl":"https://doi.org/10.1007/s00022-023-00689-z","url":null,"abstract":"","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135344559","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}
{"title":"An analytic proof of a theorem of Grace","authors":"Hiroshi Maehara, Horst Martini","doi":"10.1007/s00022-023-00691-5","DOIUrl":"https://doi.org/10.1007/s00022-023-00691-5","url":null,"abstract":"","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136015942","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}
{"title":"Equilateral triangles have minimal area and perimeter among all triangles containing a given circle in Hilbert planes","authors":"A. Nazaryan","doi":"10.1007/s00022-023-00688-0","DOIUrl":"https://doi.org/10.1007/s00022-023-00688-0","url":null,"abstract":"","PeriodicalId":15854,"journal":{"name":"Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45707698","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}