Generators and Lefschetz properties of general hyperplane arrangement apolar algebras

Abstract

Hyperplane arrangements are rich combinatorial, topological, and geometric objects with numerous applications. Apolarity is a method of turning a homogeneous polynomial into a finite graded Gorenstein algebra. Invariants of the apolar algebra of the defining equation of a hyperplane arrangement are invariants of the arrangement itself. We investigate such invariants, including the Hilbert function, minimal generators of the apolar ideal, and Lefschetz properties. The main focus of the article is on combinatorially general arrangements of degree d in dimension n with d between n and 2n-2. In this range, we determine the Hilbert function and conjecture that the same formula is valid for larger values of d too. Minimal generators of the apolar ideal in the initial degree are in natural bijection to the lines of the arrangement. The initial degree characterizes arrangements that are combinatorially general. Finally, we prove that the apolar algebra has the Weak Lefschetz Property and precisely characterize the Weak Lefschetz elements.

Lefschetz Properties of complete intersesctions and maximal variation of hyperplane sections
Mats Boij, Juan C. Migliore, Rosa M. Miró-Roig +1
arXiv18 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Weak and strong Lefschetz properties for vertex cover Artinian algebras associated to graphs
Trung Chau, Tran Quang Hoa, Dang Thi Kieu Uyen
arXiv29 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
The E₆ Restricted Hyperplane Arrangement and its E₇ Shadow: Weyl Transport on a Minuscule Bruhat Poset
Saber Ahmed, Mboyo Esole
arXiv9 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
On the generators of separable unital C*-algebras
George A. Elliott, Chun Guang Li, Zhuang Niu
arXiv22 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
Lefschetz properties for monomial complete intersections
Annet Kyomuhangi, Emanuela Marangone, Claudiu Raicu +1
arXiv8 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{buczynski2026generators,
  title = {{Generators and Lefschetz properties of general hyperplane arrangement apolar algebras}},
  author = {Jarosław Buczyński and Zach Teitler and Alexander Woo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34517},
  eprint = {2610.34517},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34517},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Generators and Lefschetz properties of general hyperplane arrangement apolar algebras
AU  - Jarosław Buczyński
AU  - Zach Teitler
AU  - Alexander Woo
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34517
UR  - https://arxiv.org/abs/2610.34517
AB  - Hyperplane arrangements are rich combinatorial, topological, and geometric objects with numerous applications. Apolarity is a method of turning a homogeneous polynomial into a finite graded Gorenstein algebra. Invariants of the apolar algebra of the defining equation of a hyperplane arrangement are invariants of the arrangement itself. We investigate such invariants, including the Hilbert function, minimal generators of the apolar ideal, and Lefschetz properties. The main focus of the article is on combinatorially general arrangements of degree d in dimension n with d between n and 2n-2. In this range, we determine the Hilbert function and conjecture that the same formula is valid for larger values of d too. Minimal generators of the apolar ideal in the initial degree are in natural bijection to the lines of the arrangement. The initial degree characterizes arrangements that are combinatorially general. Finally, we prove that the apolar algebra has the Weak Lefschetz Property and precisely characterize the Weak Lefschetz elements.
ER  -

Abstract

Related

Cite