Regularity, depth and Serre depth of symbolic powers of complementary edge ideals

@misc{ficarra2026regularity,
  title = {{Regularity, depth and Serre depth of symbolic powers of complementary edge ideals}},
  author = {Antonino Ficarra and Somayeh Moradi and Yuji Muta},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23224},
  eprint = {2610.23224},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23224},
}

TY  - UNPB
TI  - Regularity, depth and Serre depth of symbolic powers of complementary edge ideals
AU  - Antonino Ficarra
AU  - Somayeh Moradi
AU  - Yuji Muta
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23224
UR  - https://arxiv.org/abs/2610.23224
AB  - Let G be a finite simple graph and let I_c(G) denote its complementary edge ideal. In this paper, we study the Castelnuovo-Mumford regularity, the depth and Serre depth for (S_2) of the symbolic powers of I_c(G). We derive explicit formulas for reg I_c(G)^(k) and reg I_c(G)^(k)/I_c(G)^k for every k≥1, in terms of the number and the sizes of the connected components of G. As applications, we determine their asymptotic behavior, and completely characterize the symbolic powers of complementary edge ideals having linear resolutions. We then investigate the symbolic depth function k↦depth S/I_c(G)^(k). We determine explicitly its values for k=2,3,4, and establish bounds for the asymptotic depth. More generally, for an arbitrary squarefree monomial ideal I, we compare the local cohomology modules of S/I^(k) for different symbolic powers, and derive corresponding inequalities for the depth and the Serre depth. Applying this to complementary edge ideals, we obtain depth S/I_c(G)^(t)≥depth S/I_c(G)^(k) for k≥ 2t−1, despite the fact that the symbolic depth function need not be non-increasing. Finally, we characterize complementary edge ideals satisfying Serre's condition (S_r), and determine the Serre depth for (S_2) of all symbolic powers of complementary edge ideals. In particular, we obtain combinatorial characterizations of Serre's condition (S_2) and of the Cohen-Macaulayness of S/I_c(G)^(k) for every k≥ 1.
ER  -