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PreprintCC BY 4.0arXiv

Simplicial arrangements in real projective three-space revisited

  • Marek Janasz
  • Piotr Pokora
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Abstract

In this paper we study irreducible simplicial arrangements of projective planes in ℙ3(ℝ) from combinatorial and projective geometry viewpoints. We first formulate a simpliciality criterion in terms of incidences between rank-two and rank-three flats, together with equivalent formulations using face numbers, reduced restrictions, and characteristic polynomials. We also relate the classical planar restriction data of Grünbaum-Shephard to Ziegler multirestrictions. Our principal result concerns the rank-four special-vertex property: among the irreducible crystallographic Coxeter arrangements of rank four, the arrangements of types A4 and B4 admit a special vertex, whereas those of types D4 and F4 do not. A simplicial deletion chain inside B4 supplies further irreducible examples with a special vertex. Finally, we compare rank-flat, Purdy-type, and Grünbaum-Shephard defects for these arrangements.

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Cite

BibTeXRIS
BibTeX
@misc{janasz2026simplicial,
  title = {{Simplicial arrangements in real projective three-space revisited}},
  author = {Marek Janasz and Piotr Pokora},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28254},
  eprint = {2608.28254},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28254},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Simplicial arrangements in real projective three-space revisited
AU  - Marek Janasz
AU  - Piotr Pokora
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28254
UR  - https://arxiv.org/abs/2608.28254
AB  - In this paper we study irreducible simplicial arrangements of projective planes in ℙ^3(ℝ) from combinatorial and projective geometry viewpoints. We first formulate a simpliciality criterion in terms of incidences between rank-two and rank-three flats, together with equivalent formulations using face numbers, reduced restrictions, and characteristic polynomials. We also relate the classical planar restriction data of Grünbaum-Shephard to Ziegler multirestrictions. Our principal result concerns the rank-four special-vertex property: among the irreducible crystallographic Coxeter arrangements of rank four, the arrangements of types A_4 and B_4 admit a special vertex, whereas those of types D_4 and F_4 do not. A simplicial deletion chain inside B_4 supplies further irreducible examples with a special vertex. Finally, we compare rank-flat, Purdy-type, and Grünbaum-Shephard defects for these arrangements.
ER  -