Étale cohomological stability of the moduli space of stable elliptic surfaces

Banerjee,  Oishee; Park, Jun-Yong; Schmitt,  Johannes

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Étale cohomological stability of the moduli space of stable elliptic surfaces

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Park, Jun-Yong
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.48550/arXiv.2207.02496
(Preprint)

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2207.02496v1
(Preprint), 433KB

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Citation

Banerjee, O., Park, J.-Y., & Schmitt, J. (submitted). Étale cohomological stability of the moduli space of stable elliptic surfaces.

Cite as: https://hdl.handle.net/21.11116/0000-000F-C3D0-9

Abstract

We compute the (stable) étale cohomology of Homn(C,P(λ⃗ )), the moduli stack of degree n morphisms from a smooth projective curve C to the weighted projective stack P(λ⃗ ), the latter being a stacky quotient defined by P(λ⃗ ):=[AN−{0}/Gm], where Gm acts by weights λ⃗ =(λ0,⋯,λN)∈ZN+. 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.