<p>We give explicit, highly symmetric equations for the versal deformation of the singularity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_{n+1}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation> consisting of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> lines through the origin in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {A}^{n}(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">A</mi> </mrow> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in generic position. These make evident that the base space of the versal deformation of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_{n+1}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation> is isomorphic to the total space for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_{n}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mrow> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation>, if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. By induction it follows that the base space is irreducible and Gorenstein. We discuss the known connection to a modular compactification of the moduli space of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(n+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-pointed curves of genus 1. For other elliptic partition curves it seems unfeasible to compute the versal deformation in general. It is doubtful whether the base space is Gorenstein. For rational partition curves we show that the base space in general has components of different dimensions.</p>

The versal deformation of elliptic m-fold point curve singularities

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2025年9月5日

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Jan Stevens

摘要

We give explicit, highly symmetric equations for the versal deformation of the singularity $L_{n+1}^n$ $L_{n + 1}^{n}$ consisting of $n+1$ $n + 1$ lines through the origin in $\mathbb {A}^{n}(k)$ $A^{n} (k)$ in generic position. These make evident that the base space of the versal deformation of $L_{n+1}^n$ $L_{n + 1}^{n}$ is isomorphic to the total space for $L_{n}^{n-1}$ $L_{n}^{n - 1}$ , if $n\ge 5$ $n \geq 5$ . By induction it follows that the base space is irreducible and Gorenstein. We discuss the known connection to a modular compactification of the moduli space of $(n+1)$ $(n + 1)$ -pointed curves of genus 1. For other elliptic partition curves it seems unfeasible to compute the versal deformation in general. It is doubtful whether the base space is Gorenstein. For rational partition curves we show that the base space in general has components of different dimensions.