{"title":"On the conjugacy problem for finite groups in the plane Cremona group","authors":"Ilya Karzhemanov","doi":"10.1016/j.jalgebra.2024.09.034","DOIUrl":null,"url":null,"abstract":"<div><div>We give a geometric condition for two finite subgroups <span><math><mi>G</mi><mo>≃</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of the plane Cremona group <span><math><msub><mrow><mi>Cr</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> to be conjugate. The argument is based on the properties of a “Burnside-type” ring <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>G</mi><mo>)</mo></math></span> encoding rational <em>G</em>-surfaces.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"664 ","pages":"Pages 595-603"},"PeriodicalIF":0.8000,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021869324005490","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We give a geometric condition for two finite subgroups of the plane Cremona group to be conjugate. The argument is based on the properties of a “Burnside-type” ring encoding rational G-surfaces.
期刊介绍：
The Journal of Algebra is a leading international journal and publishes papers that demonstrate high quality research results in algebra and related computational aspects. Only the very best and most interesting papers are to be considered for publication in the journal. With this in mind, it is important that the contribution offer a substantial result that will have a lasting effect upon the field. The journal also seeks work that presents innovative techniques that offer promising results for future research.
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