模数堆栈 $${{\mathcal {M}}_2$$ 和 $$\overline{{{\mathcal {M}}}}_2$$ 的理论

Pub Date : 2024-07-05 DOI:10.1007/s00229-024-01581-z
Dan Edidin, Zhengning Hu
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引用次数: 0

摘要

我们分别计算了属二的平滑曲线和稳定曲线的模堆积的积分格罗登第克环(\({\mathcal {M}}}_2\), \(\overline{{\mathcal {M}}}}_2\) )。我们通过使用 Vistoli(Invent Math 131(3):635-644,1998)给出的 \({{\mathcal {M}}}_2\) 作为全局商栈的呈现来计算 \({{\textrm{K}\,}}_0({{\mathcal {M}}}_2)\) 。为了计算格罗内迪克环({{\,\textrm{K}}\、}}_0(\overline{{\mathcal {M}}}}_2)\) 我们将 \(\overline{{\mathcal {M}}}}_2\) 分解为 \(\Delta _1\) 及其补集 \(\overline{{\mathcal {M}}}}_2 \setminus \Delta _1/\),并使用 Larson (Algebr Geom 8 (3):286-318, 2021)来计算格罗滕迪克环。我们证明它们是无扭转的,这一点加上黎曼-罗赫同构让我们最终给出了积分格罗内迪克环的({{\,\textrm{K}\,}}_0(\overline{{\{mathcal {M}}}}_2)\) 呈现。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

The $${{\,\textrm{K}\,}}$$ -theory of the moduli stacks $${{\mathcal {M}}}_2$$ and $$\overline{{{\mathcal {M}}}}_2$$

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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 \({{\,\textrm{K}\,}}_0(\overline{{{\mathcal {M}}}}_2)\).

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