皮卡等级 2 的光滑射影环状变种对角线的简短解析

IF 0.9 1区数学 Q2 MATHEMATICS

Algebra & Number Theory Pub Date : 2024-10-07 DOI:10.2140/ant.2024.18.1923

Michael K. Brown, Mahrud Sayrafi

{"title":"皮卡等级 2 的光滑射影环状变种对角线的简短解析","authors":"Michael K. Brown, Mahrud Sayrafi","doi":"10.2140/ant.2024.18.1923","DOIUrl":null,"url":null,"abstract":"<p>Given a smooth projective toric variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> of Picard rank 2, we resolve the diagonal sheaf on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi>\n<mo>×</mo>\n<mi>X</mi></math> by a linear complex of length <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> dim</mi><mo> ⁡ </mo><mi>X</mi></math> consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Ermana and Smith that predicts a version of Hilbert’s syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud, Erman and Schreyer. We also apply our results to give a new proof, in the case of smooth projective toric varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier dimension of derived categories. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2\",\"authors\":\"Michael K. Brown, Mahrud Sayrafi\",\"doi\":\"10.2140/ant.2024.18.1923\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given a smooth projective toric variety <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>X</mi></math> of Picard rank 2, we resolve the diagonal sheaf on <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>X</mi>\\n<mo>×</mo>\\n<mi>X</mi></math> by a linear complex of length <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi> dim</mi><mo> ⁡ </mo><mi>X</mi></math> consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Ermana and Smith that predicts a version of Hilbert’s syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud, Erman and Schreyer. We also apply our results to give a new proof, in the case of smooth projective toric varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier dimension of derived categories. </p>\",\"PeriodicalId\":50828,\"journal\":{\"name\":\"Algebra & Number Theory\",\"volume\":\"12 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-10-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebra & Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2140/ant.2024.18.1923\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra & Number Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/ant.2024.18.1923","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

给定皮卡秩为 2 的光滑射影环 variety X，我们用长度为 dim X 的线性复数解析 X×X 上的对角剪，该复数由线束的有限直接和组成。作为应用，我们证明了贝克斯奇、埃尔马纳和史密斯猜想的一个新案例，该猜想预言了希尔伯特关于虚解析的syzygy定理的一个版本，我们还得到了皮卡等级为2的光滑投影环素上的向量束的霍罗克斯型分裂准则，扩展了艾森布德、埃尔马纳和施雷尔的一个结果。我们还应用我们的结果，在皮卡等级 2 的光滑射影环状变种的情况下，给出了奥洛夫关于派生范畴的鲁基尔维度猜想的新证明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

Given a smooth projective toric variety $X$ of Picard rank 2, we resolve the diagonal sheaf on $X \times X$ by a linear complex of length $\dim X$ consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Ermana and Smith that predicts a version of Hilbert’s syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud, Erman and Schreyer. We also apply our results to give a new proof, in the case of smooth projective toric varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier dimension of derived categories.

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

Algebra & Number Theory MATHEMATICS-

CiteScore

1.80

自引率

7.70%

发文量

审稿时长

6-12 weeks

期刊介绍： ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.