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

TL;DR

This paper computes the motive of the moduli stack of rational curves on weighted projective stacks, showing it is of mixed Tate type and providing explicit class formulas, thus advancing understanding of their arithmetic and geometric properties.

Contribution

It establishes that the motive of the moduli stack of degree n rational curves on weighted projective stacks is of mixed Tate type and computes its class in the Grothendieck ring.

Findings

The motive is of mixed Tate type over suitable fields.

The class in the Grothendieck ring is explicitly given by 6^{(a+b)n+1}-6^{(a+b)n-1}.

Improves understanding of arithmetic invariants of related moduli stacks.

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.