子空间排列的奇妙模型的胞腔$\boldsymbol{\funnyA}^1$-同调
Cellular $\mathbb{A}^1$-homology of wonderful models of subspace arrangements
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中文总结 AI 辅助
该研究计算子空间排列的De Concini–Procesi奇妙模型的胞腔$\funnyA^1$-同调，通过动机爆破计算得到关键附着类，将相关同调用区间上同调表达，并给出辫排列的显式分解与示例。
中文摘要 AI 辅助
我们计算子空间排列的De Concini–Procesi奇妙模型的胞腔$\funnyA^1$-同调。对于域$k$上的一个建筑集$\funnyG$，我们将$\funnyP(\funnyG)$的胞腔$\funnyA^1$-链复形与带有Milnor–Witt系数及导出定向数据的$\funnyη$-扭嵌套集复形相等同。关键的几何输入是一个动机爆破计算：对于沿余维数为$c$的光滑中心的爆破，相关的连接类为$(c-1)_\funnyε\funnyη$，因此当$c$为奇数时该类为零，当$c$为偶数时该类为$\funnyη$。这将Rains计算中的奇偶条件替换为Milnor–Witt附着类。作为结果，在乘以$\funnyη$后存活的胞腔$\funnyA^1$-同调部分，以及在可逆化$\funnyη$后的同调，都由$\funnyG$生成格的2-可除子偏序集的区间上同调表达。对于辫排列，该条件成为分划上的奇块条件，给出了$\funnyoverline{\funnyM}_{0,N}$的胞腔$\funnyA^1$-同调的显式分解及低秩示例。
英文摘要
We compute the cellular $\mathbb{A}^1$-homology of De Concini--Procesi wonderful models of subspace arrangements. For a building set $\mathcal{G}$ over a field $k$, we identify the cellular $\mathbb{A}^1$-chain complex of $\mathbb{P}(\mathcal{G})$ with an $η$-twisted nested-set complex carrying Milnor--Witt coefficients and derived orientation data. The key geometric input is a motivic blow-up calculation: for a blow-up along a smooth center of codimension $c$, the relevant connecting class is $(c-1)_εη$, hence it is zero for $c$ odd and $η$ for $c$ even. This replaces the parity condition in the computation of Rains by a Milnor--Witt attaching class. As a consequence, the part of cellular $\mathbb{A}^1$-homology surviving after multiplication by $η$, and also the homology after inverting $η$, are expressed by the interval cohomology of the $2$-divisible subposet of the lattice generated by $\mathcal{G}$. For the braid arrangement, the condition becomes the odd-block condition on partitions, yielding explicit decompositions for the cellular $\mathbb{A}^1$-homology of $\overline{\mathcal M}_{0,N}$ and examples in low rank.
