Nice Partitions, Supersolvability, and Freeness in Deformations of Graphic Arrangements

@misc{liang2026nice,
  title = {{Nice Partitions, Supersolvability, and Freeness in Deformations of Graphic Arrangements}},
  author = {Weikang Liang and Suijie Wang and Yue Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39407},
  eprint = {2609.39407},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39407},
}

TY  - UNPB
TI  - Nice Partitions, Supersolvability, and Freeness in Deformations of Graphic Arrangements
AU  - Weikang Liang
AU  - Suijie Wang
AU  - Yue Zhou
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39407
UR  - https://arxiv.org/abs/2609.39407
AB  - We study affine deformations 𝒜(G_𝒮) of graphic arrangements and their cones. Here G=([n],E(G)) is a simple graph, 𝒮=(S_ij) is a family of finite gain sets, and 𝒜(G_𝒮) consists of the hyperplanes x_i−x_j=a with a∈ S_ij. We call G_𝒮 blockwise admissible if every block has a vertex ordering v_1,…,v_m satisfying S_v_kv_j−S_v_kv_i⊆ S_v_iv_j for every k and all distinct i,j>k. Every such ordering is a perfect elimination ordering. We prove, for arbitrary G, that the following are equivalent: (i) G_𝒮 is blockwise admissible; (ii) the cone c𝒜(G_𝒮) is supersolvable; and (iii) 𝒜(G_𝒮) admits a nice partition. We give a direct arrangement-theoretic proof of these equivalences. For a block, the equivalence between admissibility and supersolvability is already contained in Zaslavsky's characterization of supersolvable graphic-lift lattices. We further show that every nice partition on a block is induced by an admissible ordering and that its induced edge classes are stars with distinct centers. Under these equivalent conditions, we construct a maximal modular chain through the hyperplane at infinity. We also prove that if c𝒜(G_𝒮) is free, then G is chordal.
ER  -