Let k be a characteristic zero field. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathscr {C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> be an integral affine plane k-curve. In this article, we show that the dual morphism <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Psi ^3_{{\mathcal {O}}(\mathscr {C})}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msubsup> </math></EquationSource> </InlineEquation> of the canonical morphism of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {O}}({\mathscr {C}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-modules (introduced in Le Dréau and Sebag in Osaka J Math 61(3):381–390, 2024) <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi ^3_{{\mathcal {O}}(\mathscr {C})}:\Omega _{{\mathcal {O}}(\mathscr {C})/k}^3\rightarrow {\mathcal {W}}^3_{\mathcal {O}(\mathscr {C})}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Φ</mi> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msubsup> <mo>:</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>k</mi> </mrow> <mn>3</mn> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mrow> <mi mathvariant="script">W</mi> </mrow> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, defined from the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathcal {O}}({\mathscr {C}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-module of the third-order differential forms on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\mathscr {C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> to the third-degree component of the weight grading of the k-algebra <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {O}(\mathscr {L}_{\infty }({\mathscr {C}}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">L</mi> <mi>∞</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the associated arc scheme, is injective. We describe its image and prove that this morphism <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Psi ^3_{\mathcal {O}(\mathscr {C})}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msubsup> </math></EquationSource> </InlineEquation> is not surjective in general. However, we show that the surjectivity can occur.

Weight grading and higher-order derivations on the arc scheme of an affine plane curve

期刊论文
发表日期：

2026年5月19日

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Julien Sebag

摘要

Let k be a characteristic zero field. Let ${\mathscr {C}}$ $C$ be an integral affine plane k-curve. In this article, we show that the dual morphism $\Psi ^3_{{\mathcal {O}}(\mathscr {C})}$ $Ψ_{O (C)}^{3}$ of the canonical morphism of ${\mathcal {O}}({\mathscr {C}})$ $O (C)$ -modules (introduced in Le Dréau and Sebag in Osaka J Math 61(3):381–390, 2024) $\Phi ^3_{{\mathcal {O}}(\mathscr {C})}:\Omega _{{\mathcal {O}}(\mathscr {C})/k}^3\rightarrow {\mathcal {W}}^3_{\mathcal {O}(\mathscr {C})}$ $Φ_{O (C)}^{3} : Ω_{O (C) / k}^{3} \to W_{O (C)}^{3}$ , defined from the ${\mathcal {O}}({\mathscr {C}})$ $O (C)$ -module of the third-order differential forms on ${\mathscr {C}}$ $C$ to the third-degree component of the weight grading of the k-algebra $\mathcal {O}(\mathscr {L}_{\infty }({\mathscr {C}}))$ $O (L_{\infty} (C))$ of the associated arc scheme, is injective. We describe its image and prove that this morphism $\Psi ^3_{\mathcal {O}(\mathscr {C})}$ $Ψ_{O (C)}^{3}$ is not surjective in general. However, we show that the surjectivity can occur.