{"title":"奇异扇形三次方反规范体积的有效上界","authors":"Chen Jiang, Yu Zou","doi":"10.1017/s1474748024000070","DOIUrl":null,"url":null,"abstract":"<p>For a real number <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$0<\\epsilon <1/3$</span></span></img></span></span>, we show that the anti-canonical volume of an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\epsilon $</span></span></img></span></span>-klt Fano <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$3$</span></span></img></span></span>-fold is at most <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$3,200/\\epsilon ^4$</span></span></img></span></span>, and the order <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$O(1/\\epsilon ^4)$</span></span></img></span></span> is sharp.</p>","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"22 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"AN EFFECTIVE UPPER BOUND FOR ANTI-CANONICAL VOLUMES OF SINGULAR FANO THREEFOLDS\",\"authors\":\"Chen Jiang, Yu Zou\",\"doi\":\"10.1017/s1474748024000070\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For a real number <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$0<\\\\epsilon <1/3$</span></span></img></span></span>, we show that the anti-canonical volume of an <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\epsilon $</span></span></img></span></span>-klt Fano <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$3$</span></span></img></span></span>-fold is at most <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$3,200/\\\\epsilon ^4$</span></span></img></span></span>, and the order <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307144005320-0828:S1474748024000070:S1474748024000070_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$O(1/\\\\epsilon ^4)$</span></span></img></span></span> is sharp.</p>\",\"PeriodicalId\":50002,\"journal\":{\"name\":\"Journal of the Institute of Mathematics of Jussieu\",\"volume\":\"22 1\",\"pages\":\"\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2024-03-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the Institute of Mathematics of Jussieu\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/s1474748024000070\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Institute of Mathematics of Jussieu","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s1474748024000070","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
AN EFFECTIVE UPPER BOUND FOR ANTI-CANONICAL VOLUMES OF SINGULAR FANO THREEFOLDS
For a real number $0<\epsilon <1/3$, we show that the anti-canonical volume of an $\epsilon $-klt Fano $3$-fold is at most $3,200/\epsilon ^4$, and the order $O(1/\epsilon ^4)$ is sharp.
期刊介绍：
The Journal of the Institute of Mathematics of Jussieu publishes original research papers in any branch of pure mathematics; papers in logic and applied mathematics will also be considered, particularly when they have direct connections with pure mathematics. Its policy is to feature a wide variety of research areas and it welcomes the submission of papers from all parts of the world. Selection for publication is on the basis of reports from specialist referees commissioned by the Editors.
$0<\epsilon <1/3$, we show that the anti-canonical volume of an 
$\epsilon $-klt Fano 
$3$-fold is at most 
$3,200/\epsilon ^4$, and the order 
$O(1/\epsilon ^4)$ is sharp.
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