\'Etale cohomological stability of the moduli space of stable elliptic surfaces

TL;DR

This paper computes the stable étale cohomology of moduli stacks of morphisms from curves to weighted projective stacks, using étale cohomological descent, and applies results to arithmetic conjectures over function fields.

Contribution

It introduces a method for étale cohomological descent over symmetric simplicial categories and applies it to compute cohomology of moduli stacks, resolving a geometric conjecture.

Findings

Computed stable étale cohomology of moduli stacks of morphisms.

Established étale cohomological descent over the symmetric simplicial category.

Resolved a Batyrev–Manin type conjecture for weighted projective stacks over global function fields.

Abstract

We compute the (stable) \'etale 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 \'etale 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.