{"title":"Galois points and Cremona transformations","authors":"Ahmed Abouelsaad","doi":"10.1017/s0017089524000090","DOIUrl":null,"url":null,"abstract":"In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline1.png\" /> <jats:tex-math> $\\mathrm{Bir}(\\mathbb{P}^2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove that if the Galois group has order at most <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline2.png\" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, it always extends to a subgroup of the Jonquières group associated with the point <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline3.png\" /> <jats:tex-math> $P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Conversely, with a degree of at least <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline4.png\" /> <jats:tex-math> $4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonquières maps with respect to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000090_inline5.png\" /> <jats:tex-math> $P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In addition, we also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"115 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Glasgow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0017089524000090","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to $\mathrm{Bir}(\mathbb{P}^2)$ . We prove that if the Galois group has order at most $3$ , it always extends to a subgroup of the Jonquières group associated with the point $P$ . Conversely, with a degree of at least $4$ , we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonquières maps with respect to $P$ . In addition, we also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.
期刊介绍:
Glasgow Mathematical Journal publishes original research papers in any branch of pure and applied mathematics. An international journal, its policy is to feature a wide variety of research areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics.
The journal has a web-based submission system for articles. For details of how to to upload your paper see GMJ - Online Submission Guidelines or go directly to the submission site.