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

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