图排列的向量场与单纯偏序集的面环
Vector fields of graphic arrangements and face rings of simplicial posets
AI总结：
该研究针对连通简单图对应的图排列，将其对数向量场经微小修改后与单纯偏序集的面环建立同构，进而推导其代数不变量公式并给出显式向量空间基。
AI中文摘要：
与简单图G相关联的图排列$\boldsymbol{\textit{A}}_G$是超平面排列理论中经典且被深入研究的对象。在本注记中，我们证明，对于连通图G，对$\boldsymbol{\textit{A}}_G$的对数向量场$D(\boldsymbol{\textit{A}}_G)$作微小修改后，其与某个单纯偏序集的面环同构。这使我们能根据对应单纯偏序集的组合与拓扑信息，给出$D(\boldsymbol{\textit{A}}_G)$的若干代数不变量公式，包括其希尔伯特级数、局部上同调、投射维数及Castelnuovo–Mumford正则性。作为附带结果，我们还给出了$D(\boldsymbol{\textit{A}}_G)$的显式向量空间基。
英文摘要：
A graphic arrangement $\A_G$ associated with a simple graph $G$ is a classical and well-studied object in the theory of hyperplane arrangements. In this note, we show that, for a connected graph $G$, a slight modification of the logarithmic vector field $D(\A_G)$ of $\A_G$ is isomorphic to the face ring of a certain simplicial poset. This allows us to give formulas for several algebraic invariants of $D(\A_G)$, such as its Hilbert series, local cohomology, projective dimension, and Castelnuovo--Mumford regularity, in terms of combinatorial and topological information about the corresponding simplicial poset. As a by-product, we also give an explicit vector space basis of $D(\A_G)$.
