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Abstract <p> Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8539_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource> </InlineEquation> be the variety of flexes of plane cubics. We prove that (1) <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8539_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource> </InlineEquation> is an irreducible rational algebraic variety endowed with a faithful algebraic action of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8539_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm{PSL}_3\)</EquationSource> </InlineEquation>, and (2) <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8539_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource> </InlineEquation> is <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8539_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm{PSL}_3\)</EquationSource> </InlineEquation>-equivariantly birationally isomorphic to a homogeneous fiber space over <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8539_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm{PSL}_3/K\)</EquationSource> </InlineEquation> with fiber <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8539_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb P^1\)</EquationSource> </InlineEquation> for some subgroup <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8539_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(K\)</EquationSource> </InlineEquation> isomorphic to the binary tetrahedral group <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11501_2025_8539_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm{SL}_2(\mathbb F_3)\)</EquationSource> </InlineEquation>. </p>

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The Variety of Flexes of Plane Cubics

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • Vladimir L. Popov

摘要

Abstract

Let \(X\) be the variety of flexes of plane cubics. We prove that (1) \(X\) is an irreducible rational algebraic variety endowed with a faithful algebraic action of \(\mathrm{PSL}_3\) , and (2) \(X\) is \(\mathrm{PSL}_3\) -equivariantly birationally isomorphic to a homogeneous fiber space over \(\mathrm{PSL}_3/K\) with fiber \(\mathbb P^1\) for some subgroup \(K\) isomorphic to the binary tetrahedral group \(\mathrm{SL}_2(\mathbb F_3)\) .