{"title":"实代数品种的半规范化概念","authors":"François Bernard","doi":"10.1112/jlms.12891","DOIUrl":null,"url":null,"abstract":"<p>The seminormalization of an algebraic variety <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> is the biggest variety linked to <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> by a finite, birational, and bijective morphism. In this paper, we introduce a variant of the seminormalization, suited for real algebraic varieties, called the R-seminormalization. This object has a universal property of the same kind as the one of the seminormalization but related to the real closed points of the variety. In a previous paper, the author studied the seminormalization of complex algebraic varieties using rational functions that extend continuously to the closed points for the Euclidean topology. We adapt some of those results here to the R-seminormalization, and we provide several examples. We also show that the R-seminormalization modifies the singularities of a real variety by normalizing the purely complex points and seminormalizing the real ones.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12891","citationCount":"0","resultStr":"{\"title\":\"A notion of seminormalization for real algebraic varieties\",\"authors\":\"François Bernard\",\"doi\":\"10.1112/jlms.12891\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The seminormalization of an algebraic variety <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> is the biggest variety linked to <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> by a finite, birational, and bijective morphism. In this paper, we introduce a variant of the seminormalization, suited for real algebraic varieties, called the R-seminormalization. This object has a universal property of the same kind as the one of the seminormalization but related to the real closed points of the variety. In a previous paper, the author studied the seminormalization of complex algebraic varieties using rational functions that extend continuously to the closed points for the Euclidean topology. We adapt some of those results here to the R-seminormalization, and we provide several examples. We also show that the R-seminormalization modifies the singularities of a real variety by normalizing the purely complex points and seminormalizing the real ones.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-03-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12891\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12891\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12891","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
代数变项 X $X$ 的半正化是通过有限、双向和双射态射与 X $X$ 相连的最大变项。在本文中,我们将介绍一种适合于实代数纷的半正化变体,称为 R-半正化。这个对象具有与半正化相同的普遍性质,但与实数封闭点有关。在之前的一篇论文中,作者利用连续延伸到欧几里得拓扑闭点的有理函数研究了复代数变项的半正化。我们在此将其中一些结果应用于 R-半正化,并提供了几个例子。我们还证明,R-半正化通过对纯复数点进行归一化和对实数点进行半正化来修改实数的奇异点。
A notion of seminormalization for real algebraic varieties
The seminormalization of an algebraic variety is the biggest variety linked to by a finite, birational, and bijective morphism. In this paper, we introduce a variant of the seminormalization, suited for real algebraic varieties, called the R-seminormalization. This object has a universal property of the same kind as the one of the seminormalization but related to the real closed points of the variety. In a previous paper, the author studied the seminormalization of complex algebraic varieties using rational functions that extend continuously to the closed points for the Euclidean topology. We adapt some of those results here to the R-seminormalization, and we provide several examples. We also show that the R-seminormalization modifies the singularities of a real variety by normalizing the purely complex points and seminormalizing the real ones.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.