论文标题
稳定椭圆形表面模量空间的典型同胞稳定性
Étale cohomological stability of the moduli space of stable elliptic surfaces
论文作者
论文摘要
我们计算$ \ mathrm {hom} _ {n}（c，c，\ nathcal {p}（\vecλ））$的（稳定）étale的共同体，$ n $ n $ morphisms的模量$ n $ morphisms，来自平稳的投影型c $ c $ co $ c $ c $ co $ c $ c $ c $ c $ c $ c $ \ nath c $ \ mathcal p} l l l l l l l p} l p}（p}（p}（p）由$ \ Mathcal {p}（\vecλ）定义的stacky商：= \ left [\ mathbb {a}^n- \ \ {0 \}/\ mathbb {g} _m \ right] $ \ in \ Mathbb {z}^n _ {+} $。我们的关键要素是在类别$ΔS$（对称（半）简单类别的类别上制定并证明典型的同居下降。直接的算术后果是分辨率在全球函数字段上的加权投影堆栈的几何batyrev-manin型猜想的分辨率。在此过程中，我们还分析了平滑Deligne-Mumford堆栈中向量束的加权项目化的交集理论。
We compute the (stable) étale cohomology of $\mathrm{Hom}_{n}(C, \mathcal{P}(\vecλ))$, the moduli stack of degree $n$ morphisms from a smooth projective curve $C$ to the weighted projective stack $\mathcal{P}(\vecλ)$, the latter being a stacky quotient defined by $\mathcal{P}(\vecλ) := \left[\mathbb{A}^N-\{0\}/\mathbb{G}_m\right]$, where $\mathbb{G}_m$ acts by weights $\vecλ = (λ_0, \cdots, λ_N) \in \mathbb{Z}^N_{+}$. 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.
