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<p>Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3716_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma : X\rightarrow \Delta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">Δ</mi> </mrow> </math></EquationSource> </InlineEquation> be a 1-parameter family of 2-dimensional isolated hypersurface singularities. In this paper, we show that if the Milnor number is constant, then any semistable model, obtained from <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3716_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> after a sufficiently large base change must satisfy non trivial restrictions. Those restrictions are in terms of the dual complex, Hodge structure, and numerical invariants of the central fibre.</p>

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Some consequences of the \(\mu \)-constant condition for families of surfaces

  • 期刊论文

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  • 发表日期：

    2025年3月29日

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  • Marta Aldasoro Rosales

摘要

Let \(\sigma : X\rightarrow \Delta \) σ : X → Δ be a 1-parameter family of 2-dimensional isolated hypersurface singularities. In this paper, we show that if the Milnor number is constant, then any semistable model, obtained from \(\sigma \) σ after a sufficiently large base change must satisfy non trivial restrictions. Those restrictions are in terms of the dual complex, Hodge structure, and numerical invariants of the central fibre.