Motive of the moduli stack of rational curves on a weighted projective stack

Abstract

We show the compactly supported motive of the moduli stack of degree $n$ rational curves on the weighted projective stack $\mathcal{P}(a,b)$ is of mixed Tate type over any base field $K$ with $\text{char}(K)\nmid a,b$ and has class ${\mathbb{L}}^{(a+b)n+1}-{\mathbb{L}}^{(a+b)n-1}$ in the Grothendieck ring of stacks. In particular, this improves upon the result of [HP] regarding the arithmetic invariant of the moduli stack ${\mathcal{L}}_{1,12n}:={\mathrm{Hom}}_n({\mathbb{P}}^1,{\overline{\mathcal{M}}}_{1,1})$ of stable elliptic fibrations over ${\mathbb{P}}^1$ with $12n$ nodal singular fibers and a marked Weierstrass section.

Published in Special Issue of PIMS 2019 Workshop on Arithmetic Topology