In the present paper, we study arrangements of smooth plane conics having only nodes and tacnodes as the singularities. We provide an interesting estimation on the number of nodes and tacnodes that depends only on a linear function of the number of conics. Based on that result, we obtain a new upper bound on the number of tacnodes which turns out to be better than Miyaoka’s bound for a large enough number of conics. We also study the freeness and nearly freeness of such arrangements providing a detailed description.
{"title":"On plane conic arrangements with nodes and tacnodes","authors":"A. Dimca, Marek Janasz, Piotr Pokora","doi":"10.2140/iig.2022.19.47","DOIUrl":"https://doi.org/10.2140/iig.2022.19.47","url":null,"abstract":"In the present paper, we study arrangements of smooth plane conics having only nodes and tacnodes as the singularities. We provide an interesting estimation on the number of nodes and tacnodes that depends only on a linear function of the number of conics. Based on that result, we obtain a new upper bound on the number of tacnodes which turns out to be better than Miyaoka’s bound for a large enough number of conics. We also study the freeness and nearly freeness of such arrangements providing a detailed description.","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133520334","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":"Chamber graphs of some geometries that are almost buildings","authors":"Veronica Kelsey, P. Rowley","doi":"10.2140/IIG.2019.17.189","DOIUrl":"https://doi.org/10.2140/IIG.2019.17.189","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126303651","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":"Groups of compact 8-dimensional planes: conditions implying the Lie property","authors":"H. Salzmann","doi":"10.2140/iig.2019.17.201","DOIUrl":"https://doi.org/10.2140/iig.2019.17.201","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130975852","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}
This article studies conics and subconics of $PG(2,q^2)$ and their representation in the Andr'e/Bruck-Bose setting in $PG(4,q)$. In particular, we investigate their relationship with the transversal lines of the regular spread. The main result is to show that a conic in a tangent Baer subplane of $PG(2,q^2)$ corresponds in $PG(4,q)$ to a normal rational curve that meets the transversal lines of the regular spread. Conversely, every 3 and 4-dimensional normal rational curve in $PG(4,q)$ that meets the transversal lines of the regular spread corresponds to a conic in a tangent Baer subplane of $PG(2,q^2)$.
{"title":"Conics in Baer subplanes","authors":"S. G. Barwick, Wen-Ai Jackson, P. Wild","doi":"10.2140/IIG.2019.17.85","DOIUrl":"https://doi.org/10.2140/IIG.2019.17.85","url":null,"abstract":"This article studies conics and subconics of $PG(2,q^2)$ and their representation in the Andr'e/Bruck-Bose setting in $PG(4,q)$. In particular, we investigate their relationship with the transversal lines of the regular spread. The main result is to show that a conic in a tangent Baer subplane of $PG(2,q^2)$ corresponds in $PG(4,q)$ to a normal rational curve that meets the transversal lines of the regular spread. Conversely, every 3 and 4-dimensional normal rational curve in $PG(4,q)$ that meets the transversal lines of the regular spread corresponds to a conic in a tangent Baer subplane of $PG(2,q^2)$.","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"83 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114333868","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":"A new family of 2-dimensional Laguerre planes\u0000that admit PSL2(ℝ) × ℝ as a group of automorphisms","authors":"Günter F. Steinke","doi":"10.2140/IIG.2019.17.53","DOIUrl":"https://doi.org/10.2140/IIG.2019.17.53","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130935920","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":"Ruled quintic surfaces in PG(6,q)","authors":"S. G. Barwick","doi":"10.2140/IIG.2019.17.25","DOIUrl":"https://doi.org/10.2140/IIG.2019.17.25","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121968626","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":"Sur les R-espaces","authors":"J. Tits","doi":"10.2140/iig.2018.16.59","DOIUrl":"https://doi.org/10.2140/iig.2018.16.59","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"1998 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132476513","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":"Groupes algébriques simples sur un corps\u0000local : cohomologie galoisienne, décompositions d’Iwasawa et de Cartan","authors":"F. Bruhat, J. Tits","doi":"10.2140/IIG.2018.16.219","DOIUrl":"https://doi.org/10.2140/IIG.2018.16.219","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131389602","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":"Avatars des grands théorèmes de classification\u0000d’Élie Cartan","authors":"J. Tits","doi":"10.2140/iig.2018.16.283","DOIUrl":"https://doi.org/10.2140/iig.2018.16.283","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115209787","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":"Séminaire sur les algèbres de Banach","authors":"J. Tits","doi":"10.2140/IIG.2018.16.339","DOIUrl":"https://doi.org/10.2140/IIG.2018.16.339","url":null,"abstract":"","PeriodicalId":127937,"journal":{"name":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","volume":"258 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121300912","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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