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<p>We introduce a new set of relations in the tautological Chow rings of the moduli space of stable curves of genus <i>g</i>. These relations are obtained by computing the Poincaré-dual class of empty loci in the Hodge bundle in terms of the standard generators of the tautological rings. In particular, we use these relations to obtain a new expression for the Chern classes of the Hodge bundle. We prove that the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2024_1532_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\((g-i)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo>-</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>th Chern class of the Hodge bundle, can be expressed as a linear combination of tautological classes constructed from stable graphs with at most <i>i</i> loops. In particular, the top Chern class can be expressed with trees. This property was expected as a consequence of the DR/DZ equivalence conjecture by Buryak–Guéré–Rossi.</p>

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Computations of λ-classes via strata of differentials

  • 期刊论文

  • 发表日期：

    2025年2月3日

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  • Georgios Politopoulos,
  • Adrien Sauvaget

摘要

We introduce a new set of relations in the tautological Chow rings of the moduli space of stable curves of genus g. These relations are obtained by computing the Poincaré-dual class of empty loci in the Hodge bundle in terms of the standard generators of the tautological rings. In particular, we use these relations to obtain a new expression for the Chern classes of the Hodge bundle. We prove that the \((g-i)\) ( g - i ) th Chern class of the Hodge bundle, can be expressed as a linear combination of tautological classes constructed from stable graphs with at most i loops. In particular, the top Chern class can be expressed with trees. This property was expected as a consequence of the DR/DZ equivalence conjecture by Buryak–Guéré–Rossi.