Étale cohomological stability of the moduli space of stable elliptic surfaces

2022/07/06 by Banerjee, Oishee, Park, Jun-Yong, Schmitt, Johannes
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.02496

Abstract

We compute the (stable) étale cohomology of Hom_n(C, P(λ)), the moduli stack of degree n morphisms from a smooth projective curve C to the weighted projective stack P(λ), the latter being a stacky quotient defined by P(λ) := [\mathbbA^N-\0\/\mathbbG_m], where \mathbbG_m acts by weights λ = (λ₀, ⋯, λ_N) ∈ ℤ^N₊. Our key ingredient is formulating and proving the étale cohomological descent over the category ΔS, 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.

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