Logarithmic bounds on Fujita's conjecture

IF 1.5 1区数学 Q1 MATHEMATICS

Proceedings of the London Mathematical Society Pub Date : 2024-03-15 DOI:10.1112/plms.12591

Luca Ghidelli, Justin Lacini

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We prove bounds on Fujita's basepoint freeness conjecture that grow as <mjx-container aria-label=\"n log log left parenthesis n right parenthesis\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,10\" data-semantic-content=\"11\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"n log log left parenthesis n right parenthesis\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"1,8\" data-semantic-content=\"9,1\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,2\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/10920b12-41b3-403d-9f32-532b018628ba/plms12591-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,10\" data-semantic-content=\"11\" data-semantic-role=\"implicit\" data-semantic-speech=\"n log log left parenthesis n right parenthesis\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\">⁢</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"1,8\" data-semantic-content=\"9,1\" data-semantic-parent=\"12\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\">log</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\">⁡</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,2\" data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\">log</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\">⁡</mo><mrow data-semantic-=\"\" data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow></mrow></mrow>$n\\operatorname{log}\\operatorname{log}(n)$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, where <mjx-container aria-label=\"log\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-role=\"prefix function\" data-semantic-speech=\"log\" data-semantic-type=\"function\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/39d6c5ad-3430-46ab-a455-206a37f218b7/plms12591-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-role=\"prefix function\" data-semantic-speech=\"log\" data-semantic-type=\"function\">log</mi>$\\operatorname{log}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is the logarithm with natural base.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"78 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/plms.12591","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let

X $X$

be a smooth complex projective variety of dimension

n $n$

. We prove bounds on Fujita's basepoint freeness conjecture that grow as

n \log \log (n) $n\operatorname{log}\operatorname{log}(n)$

, where

\log $\operatorname{log}$

is the logarithm with natural base.

查看原文本刊更多论文

藤田猜想的对数界限

设 X$X$ 是维数为 n$n$ 的光滑复杂射影变化。我们证明了藤田基点自由猜想的边界，其增长为 nloglog(n)$n\operatorname{log}\operatorname{log}(n)$ ，其中 log$\operatorname{log}$ 是以自然为底的对数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Proceedings of the London Mathematical Society 数学-数学

CiteScore

2.90

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The Proceedings of the London Mathematical Society is the flagship journal of the LMS. It publishes articles of the highest quality and significance across a broad range of mathematics. There are no page length restrictions for submitted papers. The Proceedings has its own Editorial Board separate from that of the Journal, Bulletin and Transactions of the LMS.