Étale cohomological stability of the moduli space of stable elliptic surfaces

Abstract

We compute the (stable) étale cohomology of ${\mathrm{Hom}}_n(C,\mathcal{P}(\vec{\lambda }))$, 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 $\mathcal{P}(\vec{\lambda }):=\left[{\mathbb{A}}^N-\{0\}/{\mathbb{G}}_m\right]$, 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 $\Delta S$, 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.

46 pages, comments are welcome