发表机构
BCAM – Basque Center for Applied Mathematics; Ikerbasque, Basque Foundation for Science; Ilam University; Okayama University（巴斯克应用数学中心; 伊克拉巴斯科，巴斯克科学基金会; 伊利姆大学; 冈山大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对有限简单图的互补边理想，给出其符号Rees代数极小生成元的组合描述，确定其生成次数上限，还分析了圈图和完全多部图符号幂的同调不变量，证明其分量线性并得到极限深度与Waldschmidt常数。
AI 中文摘要
设G是[n]上的有限简单图，I_c(G)表示多项式环S=K[x₁,…,xₙ]中它的互补边理想。我们根据G的结构，给出符号Rees代数ℛₛ(I_c(G))=⊕_{k≥0}I_c(G)^((k))t^k的极小生成元的组合描述，并证明该代数在次数不超过6时生成。此外，我们以图论术语完全确定ℛₛ(I_c(G))的极小生成元。随后，我们更详细地研究圈图和完全多部图类的符号幂I_c(G)^((k))的同调不变量，对这些族研究符号深度函数k↦depth S/I_c(G)^((k))的行为，得到符号幂的极限深度和I_c(G)的Waldschmidt常数，进一步证明所有符号幂I_c(G)^((k))都是分量线性的。
英文摘要
Let $G$ be a finite simple graph on $[n]$ and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring $S = K[x_1,\dots,x_n]$. We give a combinatorial description, in terms of the structure of $G$, of the minimal generators of the symbolic Rees algebra $\mathcal{R}_s(I_c(G)) = \bigoplus_{k \geq 0} I_c(G)^{(k)} t^k$, and show that this algebra is generated in degree at most $6$. Moreover, we completely determine the minimal generators of $\mathcal{R}_{s}(I_{c}(G))$ in graph-theoretic terms. We then study in more detail the homological invariants of the symbolic powers $I_c(G)^{(k)}$ for the classes of cycle graphs and complete multipartite graphs. For theses families, we study the behavior of the symbolic depth function $k\mapsto\operatorname{depth} S/I_c(G)^{(k)}$, we obtain the limit depth of the symbolic powers and the Waldschmidt constant of $I_c(G)$, and further prove that all the symbolic powers $I_c(G)^{(k)}$ are componentwise linear.
Comments33 pages
