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Abstract <p> We prove that there exists a constant <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8542_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(R(n)\)</EquationSource> </InlineEquation> such that a rationally connected variety of dimension <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8542_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation> over an algebraically closed field is rational if its birational automorphism group contains a <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8542_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-subgroup of maximal rank for some prime number <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8542_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(p&gt;R(n)\)</EquationSource> </InlineEquation>. This answers a question of Prokhorov and Shramov. Some applications related to the Jordan property are discussed. </p>

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Finite \(p\)-Groups of Birational Automorphisms and Characterizations of Rational Varieties

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • Jinsong Xu

摘要

Abstract

We prove that there exists a constant \(R(n)\) such that a rationally connected variety of dimension \(n\) over an algebraically closed field is rational if its birational automorphism group contains a \(p\) -subgroup of maximal rank for some prime number \(p>R(n)\) . This answers a question of Prokhorov and Shramov. Some applications related to the Jordan property are discussed.