{"title":"模数堆栈 $${{\\mathcal {M}}_2$$ 和 $$\\overline{{{\\mathcal {M}}}}_2$$ 的理论","authors":"Dan Edidin, Zhengning Hu","doi":"10.1007/s00229-024-01581-z","DOIUrl":null,"url":null,"abstract":"<p>We compute the integral Grothendieck rings of the moduli stacks, <span>\\({{\\mathcal {M}}}_2\\)</span>, <span>\\(\\overline{{{\\mathcal {M}}}}_2\\)</span> of smooth and stable curves of genus two respectively. We compute <span>\\({{\\,\\textrm{K}\\,}}_0({{\\mathcal {M}}}_2)\\)</span> by using the presentation of <span>\\({{\\mathcal {M}}}_2\\)</span> as a global quotient stack given by Vistoli (Invent Math 131(3):635–644, 1998). To compute the Grothendieck ring <span>\\({{\\,\\textrm{K}\\,}}_0(\\overline{{{\\mathcal {M}}}}_2)\\)</span> we decompose <span>\\(\\overline{{{\\mathcal {M}}}}_2\\)</span> as <span>\\(\\Delta _1\\)</span> and its complement <span>\\(\\overline{{{\\mathcal {M}}}}_2 \\setminus \\Delta _1\\)</span> and use their presentations as quotient stacks given by Larson (Algebr Geom 8 (3):286–318, 2021) to compute the Grothendieck rings. We show that they are torsion-free and this, together with the Riemann–Roch isomorphism allows us to ultimately give a presentation for the integral Grothendieck ring <span>\\({{\\,\\textrm{K}\\,}}_0(\\overline{{{\\mathcal {M}}}}_2)\\)</span>.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The $${{\\\\,\\\\textrm{K}\\\\,}}$$ -theory of the moduli stacks $${{\\\\mathcal {M}}}_2$$ and $$\\\\overline{{{\\\\mathcal {M}}}}_2$$\",\"authors\":\"Dan Edidin, Zhengning Hu\",\"doi\":\"10.1007/s00229-024-01581-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We compute the integral Grothendieck rings of the moduli stacks, <span>\\\\({{\\\\mathcal {M}}}_2\\\\)</span>, <span>\\\\(\\\\overline{{{\\\\mathcal {M}}}}_2\\\\)</span> of smooth and stable curves of genus two respectively. We compute <span>\\\\({{\\\\,\\\\textrm{K}\\\\,}}_0({{\\\\mathcal {M}}}_2)\\\\)</span> by using the presentation of <span>\\\\({{\\\\mathcal {M}}}_2\\\\)</span> as a global quotient stack given by Vistoli (Invent Math 131(3):635–644, 1998). To compute the Grothendieck ring <span>\\\\({{\\\\,\\\\textrm{K}\\\\,}}_0(\\\\overline{{{\\\\mathcal {M}}}}_2)\\\\)</span> we decompose <span>\\\\(\\\\overline{{{\\\\mathcal {M}}}}_2\\\\)</span> as <span>\\\\(\\\\Delta _1\\\\)</span> and its complement <span>\\\\(\\\\overline{{{\\\\mathcal {M}}}}_2 \\\\setminus \\\\Delta _1\\\\)</span> and use their presentations as quotient stacks given by Larson (Algebr Geom 8 (3):286–318, 2021) to compute the Grothendieck rings. We show that they are torsion-free and this, together with the Riemann–Roch isomorphism allows us to ultimately give a presentation for the integral Grothendieck ring <span>\\\\({{\\\\,\\\\textrm{K}\\\\,}}_0(\\\\overline{{{\\\\mathcal {M}}}}_2)\\\\)</span>.\\n</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00229-024-01581-z\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00229-024-01581-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The $${{\,\textrm{K}\,}}$$ -theory of the moduli stacks $${{\mathcal {M}}}_2$$ and $$\overline{{{\mathcal {M}}}}_2$$
We compute the integral Grothendieck rings of the moduli stacks, \({{\mathcal {M}}}_2\), \(\overline{{{\mathcal {M}}}}_2\) of smooth and stable curves of genus two respectively. We compute \({{\,\textrm{K}\,}}_0({{\mathcal {M}}}_2)\) by using the presentation of \({{\mathcal {M}}}_2\) as a global quotient stack given by Vistoli (Invent Math 131(3):635–644, 1998). To compute the Grothendieck ring \({{\,\textrm{K}\,}}_0(\overline{{{\mathcal {M}}}}_2)\) we decompose \(\overline{{{\mathcal {M}}}}_2\) as \(\Delta _1\) and its complement \(\overline{{{\mathcal {M}}}}_2 \setminus \Delta _1\) and use their presentations as quotient stacks given by Larson (Algebr Geom 8 (3):286–318, 2021) to compute the Grothendieck rings. We show that they are torsion-free and this, together with the Riemann–Roch isomorphism allows us to ultimately give a presentation for the integral Grothendieck ring \({{\,\textrm{K}\,}}_0(\overline{{{\mathcal {M}}}}_2)\).