{"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":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"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\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-03-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/s1474748024000070\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s1474748024000070","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","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：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
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$0<\epsilon <1/3$, 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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