{"title":"Affine homogeneous varieties and suspensions","authors":"","doi":"10.1007/s40687-024-00438-x","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>An algebraic variety <em>X</em> is called a homogeneous variety if the automorphism group <span> <span>\\({{\\,\\textrm{Aut}\\,}}(X)\\)</span> </span> acts on <em>X</em> transitively, and a homogeneous space if there exists a transitive action of an algebraic group on <em>X</em>. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we give criteria for a Danielewski surface to be a homogeneous variety and a homogeneous space. Also, we construct affine suspensions of arbitrary dimension that are homogeneous varieties but not homogeneous spaces.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-03-22","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/10.1007/s40687-024-00438-x","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
An algebraic variety X is called a homogeneous variety if the automorphism group \({{\,\textrm{Aut}\,}}(X)\) acts on X transitively, and a homogeneous space if there exists a transitive action of an algebraic group on X. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we give criteria for a Danielewski surface to be a homogeneous variety and a homogeneous space. Also, we construct affine suspensions of arbitrary dimension that are homogeneous varieties but not homogeneous spaces.
摘要 如果自变群 \({{\,\textrm{Aut}\,}}(X)\)瞬时作用于 X，则代数簇 X 称为同质簇；如果代数群的瞬时作用存在于 X，则代数簇 X 称为同质空间。作为应用，我们给出了达尼埃莱夫斯基曲面是同质变种和同质空间的标准。此外，我们还构造了任意维度的仿射悬浮，这些悬浮是同质元，但不是同质空间。
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.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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