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

We compute the integral Grothendieck rings of the moduli stacks, $M_{2}$ , ${\overline{M}}_{2}$ of smooth and stable curves of genus two respectively. We compute $K_{0} (M_{2})$ by using the presentation of $M_{2}$ as a global quotient stack given by Vistoli. To compute the Grothendieck ring $K_{0} ({\overline{M}}_{2})$ we decompose ${\overline{M}}_{2}$ as $Δ_{1}$ and its complement ${\overline{M}}_{2} ∖ Δ_{1}$ 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 $K_{0} ({\overline{M}}_{2})$ .

The $K$ -theory of the moduli stacks $M_{2}$ and ${\overline{M}}_{2}$

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The $K$ -theory of the moduli stacks $M_{2}$ and ${\overline{M}}_{2}$

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The KK-theory of the moduli stacks M2M2​ and M‾2M2​

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The KK-theory of the moduli stacks M2M2​ and M‾2M2​

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The $K$ -theory of the moduli stacks $M_{2}$ and ${\overline{M}}_{2}$

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The $K$ -theory of the moduli stacks $M_{2}$ and ${\overline{M}}_{2}$

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