{"title":"商奇点最小模型的类群","authors":"Johannes Schmitt","doi":"10.1112/blms.13100","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mi>V</mi>\n <annotation>$V$</annotation>\n </semantics></math> be a finite-dimensional vector space over the complex numbers and let <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>⩽</mo>\n <mo>SL</mo>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$G\\leqslant \\operatorname{SL}(V)$</annotation>\n </semantics></math> be a finite group. We describe the class group of a minimal model (i.e., <span></span><math>\n <semantics>\n <mi>Q</mi>\n <annotation>$\\mathbb {Q}$</annotation>\n </semantics></math>-factorial terminalization) of the linear quotient <span></span><math>\n <semantics>\n <mrow>\n <mi>V</mi>\n <mo>/</mo>\n <mi>G</mi>\n </mrow>\n <annotation>$V/G$</annotation>\n </semantics></math>. We prove that such a class group is completely controlled by the junior elements contained in <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math>.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 9","pages":"2777-2793"},"PeriodicalIF":0.8000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13100","citationCount":"0","resultStr":"{\"title\":\"The class group of a minimal model of a quotient singularity\",\"authors\":\"Johannes Schmitt\",\"doi\":\"10.1112/blms.13100\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mi>V</mi>\\n <annotation>$V$</annotation>\\n </semantics></math> be a finite-dimensional vector space over the complex numbers and let <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n <mo>⩽</mo>\\n <mo>SL</mo>\\n <mo>(</mo>\\n <mi>V</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$G\\\\leqslant \\\\operatorname{SL}(V)$</annotation>\\n </semantics></math> be a finite group. We describe the class group of a minimal model (i.e., <span></span><math>\\n <semantics>\\n <mi>Q</mi>\\n <annotation>$\\\\mathbb {Q}$</annotation>\\n </semantics></math>-factorial terminalization) of the linear quotient <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>V</mi>\\n <mo>/</mo>\\n <mi>G</mi>\\n </mrow>\\n <annotation>$V/G$</annotation>\\n </semantics></math>. We prove that such a class group is completely controlled by the junior elements contained in <span></span><math>\\n <semantics>\\n <mi>G</mi>\\n <annotation>$G$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"56 9\",\"pages\":\"2777-2793\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13100\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.13100\",\"RegionNum\":3,\"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":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13100","RegionNum":3,"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
让 V $V$ 是复数上的有限维向量空间，让 G ⩽ SL ( V ) $G\leqslant \operatorname{SL}(V)$ 是有限群。我们将描述线性商 V / G $V/G$ 的最小模型（即 Q $\mathbb {Q}$ -因子终结）的类群。我们证明这样的类群完全由 G $G$ 中包含的初等元素控制。
The class group of a minimal model of a quotient singularity
Let be a finite-dimensional vector space over the complex numbers and let be a finite group. We describe the class group of a minimal model (i.e., -factorial terminalization) of the linear quotient . We prove that such a class group is completely controlled by the junior elements contained in .
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