Bernstein-Sato ideals for free hyperplane arrangements
Abstract
Let f=(f1,…,fr) be a complete factorization of a central hyperplane arrangement D in X=ℂn. For a monoid ideal K⊆ ℕr we study the Bernstein-Sato ideal BKf of f along K, that is, the ℂ[s]-annihilator of 𝒟X[s]fs/∑m∈ K𝒟X[s]fs+m. When D is free we compute two families of these ideals with the help of AI. For the unit shift K=⟨ ei⟩ we prove that B−eif is generated by an explicit product of linear forms indexed by the dense edges of D contained in Di. This determines all the Bernstein-Sato ideals Ba,bf=Annℂ[s]𝒟X[s]fs−a/𝒟X[s]fs−b, a≥ b, of a free arrangement, generalizing formulas of Maisonobe (2016) and Bath (2020). The main new ingredient identifies the multiplicities of the relative characteristic cycle of 𝒟X[s]fs/𝒟X[s]fs+ei along the conormal bundle of the origin with the coefficients of the Hilbert series of an Artinian complete intersection attached to a generic Ziegler restriction of D; the total multiplicity computed in Wu (2022) then forces all the resulting coefficientwise upper bounds to be equalities. For the coordinate monoid ideal K=⟨ e1,…,er⟩ we show that BKf is generated by one Euler relation for each irreducible factor of the essential quotient of D. Finally, we show that the zero locus of a Bernstein-Sato ideal along a monoid ideal need not be a finite union of translated linear subvarieties, even for a reduced free arrangement in ℂ2: for f=(x,y,x+y,x+2y) and K=⟨ 3e1,3e2⟩ we compute BKf exactly and find an irreducible quadric component. This disproves a conjecture due to Budur.
Related
- Zero loci of Bernstein-Sato ideals are not necessarily linear
- On the Multiplicative Thom-Sebastiani Property for Bernstein-Sato Polynomials of Ideals
- Bernstein--Sato ideals might not be linear
- Bernstein-Sato polynomials of generic Pfaffian ideals
- A Polynomial PDE Criterion for the −n/d Root of the Bernstein-Sato polynomial of Homogeneous Ideals
BibTeX
@misc{guo2026bernsteinsato,
title = {{Bernstein-Sato ideals for free hyperplane arrangements}},
author = {Wenzong Guo and Lei Wu and Fanghan Xiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26336},
eprint = {2609.26336},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26336},
}
RIS
TY - UNPB
TI - Bernstein-Sato ideals for free hyperplane arrangements
AU - Wenzong Guo
AU - Lei Wu
AU - Fanghan Xiang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26336
UR - https://arxiv.org/abs/2609.26336
AB - Let f=(f_1,…,f_r) be a complete factorization of a central hyperplane arrangement D in X=ℂ^n. For a monoid ideal K⊆ ℕ^r we study the Bernstein-Sato ideal B^K_f of f along K, that is, the ℂ[s]-annihilator of 𝒟_X[s]f^s/∑_m∈ K𝒟_X[s]f^s+m. When D is free we compute two families of these ideals with the help of AI. For the unit shift K=⟨ e_i⟩ we prove that B^−e_i_f is generated by an explicit product of linear forms indexed by the dense edges of D contained in D_i. This determines all the Bernstein-Sato ideals B^a,b_f=Ann_ℂ[s]𝒟_X[s]f^s−a/𝒟_X[s]f^s−b, a≥ b, of a free arrangement, generalizing formulas of Maisonobe (2016) and Bath (2020). The main new ingredient identifies the multiplicities of the relative characteristic cycle of 𝒟_X[s]f^s/𝒟_X[s]f^s+e_i along the conormal bundle of the origin with the coefficients of the Hilbert series of an Artinian complete intersection attached to a generic Ziegler restriction of D; the total multiplicity computed in Wu (2022) then forces all the resulting coefficientwise upper bounds to be equalities. For the coordinate monoid ideal K=⟨ e_1,…,e_r⟩ we show that B^K_f is generated by one Euler relation for each irreducible factor of the essential quotient of D. Finally, we show that the zero locus of a Bernstein-Sato ideal along a monoid ideal need not be a finite union of translated linear subvarieties, even for a reduced free arrangement in ℂ^2: for f=(x,y,x+y,x+2y) and K=⟨ 3e_1,3e_2⟩ we compute B^K_f exactly and find an irreducible quadric component. This disproves a conjecture due to Budur.
ER -