We study the integral Chow ring of the stack <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1641_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_{g,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> parametrizing n-pointed smooth hyperelliptic curves of genus g. We compute the integral Chow ring of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1641_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_{g,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1641_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$n=1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> completely, while for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1641_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="110" /> </InlineMediaObject> <EquationSource Format="TEX">$3\le n\le 2g+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>2</mn> <mi>g</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> we compute it up to the additive order of a single class in degree 2. We obtain partial results also for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1641_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$n=2g+3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mi>g</mi> <mo>+</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, taking <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1641_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$g=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and recalling that <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1641_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_{2,n}=\mathcal {M}_{2,n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="script">M</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>, our results hold for <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1641_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{CH}^*(\mathcal {M}_{2,n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mtext>CH</mtext> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">M</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="229_2025_1641_Article_IEq9.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le n\le 7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>.

The integral chow ring of the stack of pointed hyperelliptic curves

期刊论文
发表日期：

2025年6月1日

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Alberto Landi

摘要

We study the integral Chow ring of the stack $\mathcal {H}_{g,n}$ $H_{g, n}$ parametrizing n-pointed smooth hyperelliptic curves of genus g. We compute the integral Chow ring of $\mathcal {H}_{g,n}$ $H_{g, n}$ for $n=1,2$ $n = 1, 2$ completely, while for $3\le n\le 2g+2$ $3 \leq n \leq 2 g + 2$ we compute it up to the additive order of a single class in degree 2. We obtain partial results also for $n=2g+3$ $n = 2 g + 3$ . In particular, taking $g=2$ $g = 2$ and recalling that $\mathcal {H}_{2,n}=\mathcal {M}_{2,n}$ $H_{2, n} = M_{2, n}$ , our results hold for $\textrm{CH}^*(\mathcal {M}_{2,n})$ ${CH}^{*} (M_{2, n})$ for $1\le n\le 7$ $1 \leq n \leq 7$ .