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<p>We establish a Hilbert–Samuel formula for semiample and semipositive adelic line bundles by providing estimates on their minimal slopes. We thus show the birational invariance of the arithmetic <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_826_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation>-volume, which gives an application on how generic and small algebraic points act under a birational morphism. Additionally we give the continuous extension of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_826_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation>-volume on a semiample cone.</p>

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Continuous extension and birational invariance of \(\chi \)-volume over an adelic curve

  • 期刊论文

  • 发表日期：

    2025年5月22日

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  • Wenbin Luo

摘要

We establish a Hilbert–Samuel formula for semiample and semipositive adelic line bundles by providing estimates on their minimal slopes. We thus show the birational invariance of the arithmetic \(\chi \) χ -volume, which gives an application on how generic and small algebraic points act under a birational morphism. Additionally we give the continuous extension of \(\chi \) χ -volume on a semiample cone.