{"title":"贝尔子平面中的圆锥曲线","authors":"S. G. Barwick, Wen-Ai Jackson, P. Wild","doi":"10.2140/IIG.2019.17.85","DOIUrl":null,"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":0.0000,"publicationDate":"2019-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Conics in Baer subplanes\",\"authors\":\"S. G. Barwick, Wen-Ai Jackson, P. Wild\",\"doi\":\"10.2140/IIG.2019.17.85\",\"DOIUrl\":null,\"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\":0.0000,\"publicationDate\":\"2019-03-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2140/IIG.2019.17.85\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/IIG.2019.17.85","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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 $PG(2,q^2)$.
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