<p>The lattice cohomology of a reduced curve singularity was introduced in [<CitationRef CitationID="CR4">4</CitationRef>]. It is a bigraded <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1073_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}[U]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">[</mo> <mi>U</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>-module <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1073_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {H}}^*=\oplus _{q,n}{\mathbb {H}}^q_{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mo>∗</mo> </msup> <mo>=</mo> <msub> <mo>⊕</mo> <mrow> <mi>q</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msubsup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>q</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, that categorifies the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1073_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-invariant and extracts key geometric information from the semigroup of values.</p><p>In the present paper we prove three structure theorems for this new invariant: (a) the weight-grading of the reduced cohomology is – just as in the case of the topological lattice cohomology of normal surface singularities [<CitationRef CitationID="CR22">22</CitationRef>] – nonpositive; (b) the graded <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1073_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}[U]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">[</mo> <mi>U</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>-module structure of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1073_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {H}}^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> determines whether or not a given curve is Gorenstein; and finally (c) the lattice cohomology module <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1073_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {H}}^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> of any plane curve singularity determines its multiplicity.</p>

Structural properties of the lattice cohomology of curve singularities

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2025年8月7日

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Alexander A. Kubasch,
András Némethi,
Gergő Schefler

摘要

The lattice cohomology of a reduced curve singularity was introduced in [4]. It is a bigraded $\mathbb {Z}[U]$ $Z [U]$ -module ${\mathbb {H}}^*=\oplus _{q,n}{\mathbb {H}}^q_{2n}$ $H^{*} = \oplus_{q, n} H_{2 n}^{q}$ , that categorifies the $\delta $ $δ$ -invariant and extracts key geometric information from the semigroup of values.

In the present paper we prove three structure theorems for this new invariant: (a) the weight-grading of the reduced cohomology is – just as in the case of the topological lattice cohomology of normal surface singularities [22] – nonpositive; (b) the graded $\mathbb {Z}[U]$ $Z [U]$ -module structure of ${\mathbb {H}}^0$ $H^{0}$ determines whether or not a given curve is Gorenstein; and finally (c) the lattice cohomology module ${\mathbb {H}}^0$ $H^{0}$ of any plane curve singularity determines its multiplicity.