2022/07/06 by Oishee Banerjee, Jun-Yong Park, Banerjee, Oishee +4 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Cohomology #Combinatorics #Conjecture #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematics #Moduli space #Morphism #Number Theory (math.NT) #Pure mathematics #Quotient #Stack (abstract data type) #Vector bundle #math.AG #math.AT #math.NT
paper · pdf · doi:10.48550/arxiv.2207.02496
published in arXiv (Cornell University) (Cornell University) · 46 pages, comments are welcome
arxiv created 2022/07/06 · openalex publication_date 2022/07/06 · arxiv updated 2022/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We compute the (stable) étale cohomology of Homn(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) := [\mathbbAN-\0\/\mathbbGm], where \mathbbGm acts by weights λ = (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.