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arXiv · 2311.12122

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

Dan Edidin·Zhengning Hu
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Abstract

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 $K_0(\mathcal{M}_2)$ by using the presentation of $\mathcal{M}_2$ as a global quotient stack given by Vistoli. To compute the Grothendieck ring $K_0(\overline{\mathcal{M}}_2)$ 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 to compute their Grothendieck rings. We show that they are torsion-free and this, together with the Riemann-Roch isomorphism allows to ultimately give a presentation for the integral Grothendieck ring $K_0(\overline{\mathcal{M}}_2)$.

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Dan Edidin, Zhengning Hu. 2023-11-20. The $K$-theory of the moduli stacks $\mathcal{M}_2$ and $\overline{\mathcal{M}}_2$. https://arxiv.org/abs/2311.12122

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