{"title":"The behavior of essential dimension under specialization, II","authors":"Zinovy Reichstein, Federico Scavia","doi":"10.2140/ant.2023.17.1925","DOIUrl":null,"url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>-varieties over a base variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math>, the essential dimension of the geometric fibers may drop on a countable union of Zariski closed subsets of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math> and stays constant away from this countable union. We give several applications of this result. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra & Number Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/ant.2023.17.1925","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
Let be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free -varieties over a base variety , the essential dimension of the geometric fibers may drop on a countable union of Zariski closed subsets of and stays constant away from this countable union. We give several applications of this result.
期刊介绍：
ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms.
The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.
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