AI 中文总结
本文研究彼得森簇的几何不变量理论，利用理查森分层明确其GIT商的稳定与半稳定轨迹，确定腔分解与跨壁态射，还刻画正则性并证明商光滑的充要条件，揭示彼得森簇奇异几何与GIT商变化的关联。
AI 中文摘要
我们研究彼得森簇$\text{Pet}_n\ ext{GL}(n,\boldsymbol{C})/B$在单参数子群$\boldsymbol{\nu}:\boldsymbol{G}_m\to T$下，关于根格中正则支配特征$\boldsymbol{\nu}$给出的线性化$\boldsymbol{\nu}(\boldsymbol{\nu})$的GIT商。利用理查森分层，我们用单根的子集明确描述半稳定和稳定轨迹，确定GIT腔分解及对应的跨壁态射。在深腔中，该商同构于加权射影空间$\boldsymbol{P}(1,2,\boldsymbol{n}-1)$。我们得到正则性的完整腔理论刻画，描述正则性如何随线性化选择变化，还证明该商光滑当且仅当$n\boldsymbol{\nu}3$，与正则支配线性化无关。这些结果刻画了彼得森簇的奇异几何如何在其GIT商的变化中体现。
英文摘要
We study the GIT quotients of the Peterson variety $\mathrm{Pet}_n\subset \mathrm{GL}(n,\mathbb C)/B$ under a one-parameter subgroup $λ:\mathbb G_m \to T$ with respect to the linearization $\mathcal L(χ)$ given by a regular dominant character $χ$ in the root lattice. Using the Richardson stratification, we describe the semistable and stable loci explicitly in terms of subsets of simple roots. This determines the GIT chamber decomposition and the corresponding wall-crossing morphisms. In the deep chamber, the quotient is shown to be isomorphic to the weighted projective space $\mathbb P(1,2,\ldots,n-1)$. We obtain a complete chamber-theoretic characterization of normality and describe how normality varies with the choice of linearization. We also prove that the quotient is smooth if and only if $n\le3$, independently of the regular dominant linearization. These results describe how the singular geometry of the Peterson variety is reflected in the variation of its GIT quotients.
Comments47 pages, comments are welcome
