Étale cohomological stability of the moduli space of stable elliptic surfaces

We compute the (stable) étale cohomology of ${\mathrm{H}\mathrm{o}\mathrm{m}}_n(C,\mathcal{P}(\vec{\lambda }))$, the moduli stack of degree $n$ morphisms from a smooth projective curve $C$ to the weighted projective stack $\mathcal{P}(\vec{\lambda })$, the latter being a stacky quotient defined by $\mathcal{P}(\vec{\lambda }):=[{\mathbb{A}}^N-\{0\}\mathrm{/}{\mathbb{G}}_m]$, where ${\mathbb{G}}_m$ acts by weights $\vec{\lambda }=({\lambda }_0,\cdots ,{\lambda }_N)\in {\mathbb{Z}}_{+}^N$. Our key ingredient is formulating and proving the étale cohomological descent over the category $\mathrm{\Delta }S$, the symmetric (semi)simplicial category. An immediate arithmetic consequence is the resolution of the geometric Batyrev–Manin type conjecture for weighted projective stacks over global function fields. Along the way, we also analyze the intersection theory on weighted projectivizations of vector bundles on smooth Deligne-Mumford stacks.