拟阵舒伯特簇何时为$\boldsymbol{\funnybb{Q}}$-戈伦斯坦（$\funnybb{Q}$-Gorenstein）？
When is the matroid Schubert variety $\mathbb{Q}$-Gorenstein?
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中文总结 AI 辅助
该研究针对拟阵舒伯特簇$Y_V$，运用运算乔伊上同调证明其上线丛的限制性质，给出其为戈伦斯坦与$\funnybb{Q}$-戈伦斯坦的组合学刻画，并构造了三类不同性质的实例。
中文摘要 AI 辅助
设$E$为有限集，$V \ni \funnybb{C}^E$为不包含于任何坐标超平面的线性子空间，$V$在射影直线乘积空间$(\funnybb{P}^1)^E$中的闭包$Y_V$称为拟阵舒伯特簇（或排列舒伯特簇）。我们运用运算乔伊上同调证明，$Y_V$上的每一条线丛均为$(\funnybb{P}^1)^E$上线丛的限制；随后分别给出$Y_V$为戈伦斯坦（Gorenstein）与$\funnybb{Q}$-戈伦斯坦的组合学刻画；还提供了三类线性子空间$V$的实例：$Y_V$为戈伦斯坦但非光滑、为$\funnybb{Q}$-戈伦斯坦但非戈伦斯坦、以及非$\funnybb{Q}$-戈伦斯坦。
英文摘要
Let $E$ be a finite set, and let $V \subseteq \mathbb{C}^E$ be a linear subspace that is not contained in any coordinate hyperplane. The closure of $V$ in the product of projective lines $(\mathbb{P}^1)^E$ is a singular variety $Y_V$ known as the matroid Schubert variety (or arrangement Schubert variety). We use operational Chow cohomology to prove that every line bundle on $Y_V$ is the restriction of a line bundle on $(\mathbb{P}^1)^E.$ We then give combinatorial characterizations of when $Y_V$ is Gorenstein and $\mathbb{Q}$-Gorenstein, respectively. We provide examples of linear subspaces $V$ such that $Y_V$ is Gorenstein but not smooth, $\mathbb{Q}$-Gorenstein but not Gorenstein, and not $\mathbb{Q}$-Gorenstein, respectively.
