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, \({{\mathcal {M}}}_2\) ${\mathcal{M}}_2$ , \(\overline{{{\mathcal {M}}}}_2\) ${\overline{\mathcal{M}}}_2$ of smooth and stable curves of genus two respectively. We compute \({{\,\textrm{K}\,}}_0({{\mathcal {M}}}_2)\) ${\text{K}}_0({\mathcal{M}}_2)$ by using the presentation of \({{\mathcal {M}}}_2\) ${\mathcal{M}}_2$ as a global quotient stack given by Vistoli (Invent Math 131(3):635–644, 1998). To compute the Grothendieck ring \({{\,\textrm{K}\,}}_0(\overline{{{\mathcal {M}}}}_2)\) ${\text{K}}_0({\overline{\mathcal{M}}}_2)$ we decompose \(\overline{{{\mathcal {M}}}}_2\) ${\overline{\mathcal{M}}}_2$ as \(\Delta _1\) ${\mathrm{\Delta }}_1$ and its complement \(\overline{{{\mathcal {M}}}}_2 \setminus \Delta _1\) ${\overline{\mathcal{M}}}_2\backslash {\mathrm{\Delta }}_1$ 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 \({{\,\textrm{K}\,}}_0(\overline{{{\mathcal {M}}}}_2)\) ${\text{K}}_0({\overline{\mathcal{M}}}_2)$ .