{"title":"光滑曲线的双族性","authors":"E. Ballico","doi":"10.2478/UMCSMATH-2014-0002","DOIUrl":null,"url":null,"abstract":"Let \\(C\\) be a smooth curve of genus \\(g\\). For each positive integer \\(r\\) the birational \\(r\\)-gonality \\(s_r(C)\\) of \\(C\\) is the minimal integer \\(t\\) such that there is \\(L\\in \\mbox{Pic}^t(C)\\) with \\(h^0(C,L) =r+1\\). Fix an integer \\(r\\ge 3\\). In this paper we prove the existence of an integer \\(g_r\\) such that for every integer \\(g\\ge g_r\\) there is a smooth curve \\(C\\) of genus \\(g\\) with \\(s_{r+1}(C)/(r+1) > s_r(C)/r\\), i.e. in the sequence of all birational gonalities of \\(C\\) at least one of the slope inequalities fails.","PeriodicalId":340819,"journal":{"name":"Annales Umcs, Mathematica","volume":"32 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2014-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"On the birational gonalities of smooth curves\",\"authors\":\"E. Ballico\",\"doi\":\"10.2478/UMCSMATH-2014-0002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let \\\\(C\\\\) be a smooth curve of genus \\\\(g\\\\). For each positive integer \\\\(r\\\\) the birational \\\\(r\\\\)-gonality \\\\(s_r(C)\\\\) of \\\\(C\\\\) is the minimal integer \\\\(t\\\\) such that there is \\\\(L\\\\in \\\\mbox{Pic}^t(C)\\\\) with \\\\(h^0(C,L) =r+1\\\\). Fix an integer \\\\(r\\\\ge 3\\\\). In this paper we prove the existence of an integer \\\\(g_r\\\\) such that for every integer \\\\(g\\\\ge g_r\\\\) there is a smooth curve \\\\(C\\\\) of genus \\\\(g\\\\) with \\\\(s_{r+1}(C)/(r+1) > s_r(C)/r\\\\), i.e. in the sequence of all birational gonalities of \\\\(C\\\\) at least one of the slope inequalities fails.\",\"PeriodicalId\":340819,\"journal\":{\"name\":\"Annales Umcs, Mathematica\",\"volume\":\"32 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2014-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales Umcs, Mathematica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2478/UMCSMATH-2014-0002\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales Umcs, Mathematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/UMCSMATH-2014-0002","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Let \(C\) be a smooth curve of genus \(g\). For each positive integer \(r\) the birational \(r\)-gonality \(s_r(C)\) of \(C\) is the minimal integer \(t\) such that there is \(L\in \mbox{Pic}^t(C)\) with \(h^0(C,L) =r+1\). Fix an integer \(r\ge 3\). In this paper we prove the existence of an integer \(g_r\) such that for every integer \(g\ge g_r\) there is a smooth curve \(C\) of genus \(g\) with \(s_{r+1}(C)/(r+1) > s_r(C)/r\), i.e. in the sequence of all birational gonalities of \(C\) at least one of the slope inequalities fails.