Symbolic Rees algebras of complementary edge ideals
Abstract
Let G be a finite simple graph on [n] and let Ic(G) denote its complementary edge ideal in the polynomial ring S = K[x1,…,xn]. We give a combinatorial description, in terms of the structure of G, of the minimal generators of the symbolic Rees algebra ℛs(Ic(G)) = ⨁k ≥ 0 Ic(G)(k) tk, and show that this algebra is generated in degree at most $6$. Moreover, we completely determine the minimal generators of ℛs(Ic(G)) in graph-theoretic terms. We then study in more detail the homological invariants of the symbolic powers Ic(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↦depth S/Ic(G)(k), we obtain the limit depth of the symbolic powers and the Waldschmidt constant of Ic(G), and further prove that all the symbolic powers Ic(G)(k) are componentwise linear.
Related
- Regularity of symbolic powers of complementary edge ideals
- Regularity, depth and Serre depth of symbolic powers of complementary edge ideals
- Symbolic depth functions of edge ideals via local graph structure
- Comparisons between ordinary and symbolic powers of edge ideals with respect to regularity and depth
- Cartwright--Sturmfelsness of complementary determinantal edge ideals
BibTeX
@misc{ficarra2026symbolic,
title = {{Symbolic Rees algebras of complementary edge ideals}},
author = {Antonino Ficarra and Somayeh Moradi and Yuji Muta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01165},
eprint = {2609.01165},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01165},
}
RIS
TY - UNPB
TI - Symbolic Rees algebras of complementary edge ideals
AU - Antonino Ficarra
AU - Somayeh Moradi
AU - Yuji Muta
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.01165
UR - https://arxiv.org/abs/2609.01165
AB - 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,…,x_n]. We give a combinatorial description, in terms of the structure of G, of the minimal generators of the symbolic Rees algebra ℛ_s(I_c(G)) = ⨁_k ≥ 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 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↦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.
ER -