给定次平面曲线的最坏奇异性。

IF 1.2 2区 数学 Q1 MATHEMATICS
Journal of Geometric Analysis Pub Date : 2017-01-01 Epub Date: 2017-02-07 DOI:10.1007/s12220-017-9762-y
Ivan Cheltsov
{"title":"给定次平面曲线的最坏奇异性。","authors":"Ivan Cheltsov","doi":"10.1007/s12220-017-9762-y","DOIUrl":null,"url":null,"abstract":"<p><p>We prove that <math> <mrow><mfrac><mn>2</mn> <mi>d</mi></mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <msup><mrow><mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mn>2</mn></msup> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>1</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>5</mn></mrow> <mrow><msup><mi>d</mi> <mn>2</mn></msup> <mo>-</mo> <mn>3</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn></mrow> </mfrac> </mrow> </math> and <math> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo></mrow> </mfrac> </math> are the smallest log canonical thresholds of reduced plane curves of degree <math><mrow><mi>d</mi> <mo>⩾</mo> <mn>3</mn></mrow> </math> , and we describe reduced plane curves of degree <i>d</i> whose log canonical thresholds are these numbers. As an application, we prove that <math> <mrow><mfrac><mn>2</mn> <mi>d</mi></mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <msup><mrow><mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mn>2</mn></msup> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>1</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>5</mn></mrow> <mrow><msup><mi>d</mi> <mn>2</mn></msup> <mo>-</mo> <mn>3</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn></mrow> </mfrac> </mrow> </math> and <math> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo></mrow> </mfrac> </math> are the smallest values of the <math><mi>α</mi></math> -invariant of Tian of smooth surfaces in <math> <msup><mrow><mi>P</mi></mrow> <mn>3</mn></msup> </math> of degree <math><mrow><mi>d</mi> <mo>⩾</mo> <mn>3</mn></mrow> </math> . We also prove that every reduced plane curve of degree <math><mrow><mi>d</mi> <mo>⩾</mo> <mn>4</mn></mrow> </math> whose log canonical threshold is smaller than <math><mfrac><mn>5</mn> <mrow><mn>2</mn> <mi>d</mi></mrow> </mfrac> </math> is GIT-unstable for the action of the group <math> <mrow><msub><mi>PGL</mi> <mn>3</mn></msub> <mrow><mo>(</mo> <mi>C</mi> <mo>)</mo></mrow> </mrow> </math> , and we describe GIT-semistable reduced plane curves with log canonical thresholds  <math><mfrac><mn>5</mn> <mrow><mn>2</mn> <mi>d</mi></mrow> </mfrac> </math> .</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9762-y","citationCount":"14","resultStr":"{\"title\":\"Worst Singularities of Plane Curves of Given Degree.\",\"authors\":\"Ivan Cheltsov\",\"doi\":\"10.1007/s12220-017-9762-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>We prove that <math> <mrow><mfrac><mn>2</mn> <mi>d</mi></mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <msup><mrow><mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mn>2</mn></msup> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>1</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>5</mn></mrow> <mrow><msup><mi>d</mi> <mn>2</mn></msup> <mo>-</mo> <mn>3</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn></mrow> </mfrac> </mrow> </math> and <math> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo></mrow> </mfrac> </math> are the smallest log canonical thresholds of reduced plane curves of degree <math><mrow><mi>d</mi> <mo>⩾</mo> <mn>3</mn></mrow> </math> , and we describe reduced plane curves of degree <i>d</i> whose log canonical thresholds are these numbers. As an application, we prove that <math> <mrow><mfrac><mn>2</mn> <mi>d</mi></mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <msup><mrow><mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mn>2</mn></msup> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>1</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>5</mn></mrow> <mrow><msup><mi>d</mi> <mn>2</mn></msup> <mo>-</mo> <mn>3</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn></mrow> </mfrac> </mrow> </math> and <math> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo></mrow> </mfrac> </math> are the smallest values of the <math><mi>α</mi></math> -invariant of Tian of smooth surfaces in <math> <msup><mrow><mi>P</mi></mrow> <mn>3</mn></msup> </math> of degree <math><mrow><mi>d</mi> <mo>⩾</mo> <mn>3</mn></mrow> </math> . We also prove that every reduced plane curve of degree <math><mrow><mi>d</mi> <mo>⩾</mo> <mn>4</mn></mrow> </math> whose log canonical threshold is smaller than <math><mfrac><mn>5</mn> <mrow><mn>2</mn> <mi>d</mi></mrow> </mfrac> </math> is GIT-unstable for the action of the group <math> <mrow><msub><mi>PGL</mi> <mn>3</mn></msub> <mrow><mo>(</mo> <mi>C</mi> <mo>)</mo></mrow> </mrow> </math> , and we describe GIT-semistable reduced plane curves with log canonical thresholds  <math><mfrac><mn>5</mn> <mrow><mn>2</mn> <mi>d</mi></mrow> </mfrac> </math> .</p>\",\"PeriodicalId\":56121,\"journal\":{\"name\":\"Journal of Geometric Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2017-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s12220-017-9762-y\",\"citationCount\":\"14\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Geometric Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s12220-017-9762-y\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2017/2/7 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Geometric Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s12220-017-9762-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2017/2/7 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 14

摘要

我们证明2 d, 2 d - 3 (d - 1) 2, 2 d - 1 d (d - 1), 2 d - 5 d 2 - 3 d + 1和2 d - 3 d (d - 2)的平面曲线的最小日志规范阈值度d⩾3,我们描述了平面曲线度d的日志规范阈值这些数字。作为一个应用,我们证明了2d, 2d - 3 (d - 1) 2, 2d - 1 d (d - 1), 2d - 5 d 2 - 3 d + 1和2d - 3 d (d - 2)是度为d大于或等于3的p3光滑表面的Tian的α -不变量的最小值。我们还证明,对于PGL 3 (C)组的作用而言,其对数规范阈值小于52d的每个度d小于或等于4的简化平面曲线是git不稳定的,并且我们用对数规范阈值52d描述了git半稳定的简化平面曲线。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Worst Singularities of Plane Curves of Given Degree.

We prove that 2 d , 2 d - 3 ( d - 1 ) 2 , 2 d - 1 d ( d - 1 ) , 2 d - 5 d 2 - 3 d + 1 and 2 d - 3 d ( d - 2 ) are the smallest log canonical thresholds of reduced plane curves of degree d 3 , and we describe reduced plane curves of degree d whose log canonical thresholds are these numbers. As an application, we prove that 2 d , 2 d - 3 ( d - 1 ) 2 , 2 d - 1 d ( d - 1 ) , 2 d - 5 d 2 - 3 d + 1 and 2 d - 3 d ( d - 2 ) are the smallest values of the α -invariant of Tian of smooth surfaces in P 3 of degree d 3 . We also prove that every reduced plane curve of degree d 4 whose log canonical threshold is smaller than 5 2 d is GIT-unstable for the action of the group PGL 3 ( C ) , and we describe GIT-semistable reduced plane curves with log canonical thresholds  5 2 d .

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
2.00
自引率
9.10%
发文量
290
审稿时长
3 months
期刊介绍: JGA publishes both research and high-level expository papers in geometric analysis and its applications. There are no restrictions on page length.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信