{"title":"关于确定一条标准曲线的点的数目","authors":"Jan Stevens","doi":"10.1016/1385-7258(89)90012-7","DOIUrl":null,"url":null,"abstract":"<div><p>We show that through <em>g</em> + 5 general points of <span><math><mtext>P</mtext></math></span><sup><em>g</em>−1</sup> passes a canonical curve of genus g. For low, odd values of g one may assign more points (<em>g</em>+5+[6/<em>g</em>−2]). The proof is based on a study of the normal bundle of <em>g</em>-cuspidal rational curves.</p><p>These facts have consequences for the smoothability of the curve singularity consisting of r lines through the origin of <em>k</em><sup><em>n</em></sup>, <em>n</em><<em>r</em>≤(<sub>2</sub><sup><em>n</em>+1</sup>), in generic position. We strengthen some results of Pinkham and Greuel on such curves.</p></div>","PeriodicalId":100664,"journal":{"name":"Indagationes Mathematicae (Proceedings)","volume":"92 4","pages":"Pages 485-494"},"PeriodicalIF":0.0000,"publicationDate":"1989-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/1385-7258(89)90012-7","citationCount":"22","resultStr":"{\"title\":\"On the number of points determining a canonical curve\",\"authors\":\"Jan Stevens\",\"doi\":\"10.1016/1385-7258(89)90012-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We show that through <em>g</em> + 5 general points of <span><math><mtext>P</mtext></math></span><sup><em>g</em>−1</sup> passes a canonical curve of genus g. For low, odd values of g one may assign more points (<em>g</em>+5+[6/<em>g</em>−2]). The proof is based on a study of the normal bundle of <em>g</em>-cuspidal rational curves.</p><p>These facts have consequences for the smoothability of the curve singularity consisting of r lines through the origin of <em>k</em><sup><em>n</em></sup>, <em>n</em><<em>r</em>≤(<sub>2</sub><sup><em>n</em>+1</sup>), in generic position. We strengthen some results of Pinkham and Greuel on such curves.</p></div>\",\"PeriodicalId\":100664,\"journal\":{\"name\":\"Indagationes Mathematicae (Proceedings)\",\"volume\":\"92 4\",\"pages\":\"Pages 485-494\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1989-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/1385-7258(89)90012-7\",\"citationCount\":\"22\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Indagationes Mathematicae (Proceedings)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/1385725889900127\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae (Proceedings)","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/1385725889900127","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the number of points determining a canonical curve
We show that through g + 5 general points of g−1 passes a canonical curve of genus g. For low, odd values of g one may assign more points (g+5+[6/g−2]). The proof is based on a study of the normal bundle of g-cuspidal rational curves.
These facts have consequences for the smoothability of the curve singularity consisting of r lines through the origin of kn, n<r≤(2n+1), in generic position. We strengthen some results of Pinkham and Greuel on such curves.
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