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<p>We compute the quadratic Euler characteristic of the symmetric powers of a smooth, projective curve over any field <i>k</i> that is not of characteristic two, using the Motivic Gauss–Bonnet Theorem of Levine–Raksit. As an application, we show that the power structure on the Grothendieck–Witt ring introduced by Pajwani–Pál computes the compactly supported <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1623_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {A}}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">A</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-Euler characteristic of symmetric powers for all curves.</p>

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Quadratic Euler characteristic of symmetric powers of curves

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月20日

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  • Lukas F. Bröring,
  • Anna M. Viergever

摘要

We compute the quadratic Euler characteristic of the symmetric powers of a smooth, projective curve over any field k that is not of characteristic two, using the Motivic Gauss–Bonnet Theorem of Levine–Raksit. As an application, we show that the power structure on the Grothendieck–Witt ring introduced by Pajwani–Pál computes the compactly supported \({\mathbb {A}}^1\) A 1 -Euler characteristic of symmetric powers for all curves.