Combinatorially Determined Zeroes of Bernstein--Sato Ideals for Tame and Free Arrangements

TL;DR

This paper computes the zero loci and roots of Bernstein--Sato ideals for hyperplane arrangements, especially free and tame ones, providing combinatorial formulas and extending duality results to understand divisor freeness.

Contribution

It generalizes techniques to compute Bernstein--Sato ideals for various arrangements, including non-reduced and non-free cases, and links roots to hyperplane arrangement properties.

Findings

Computed zero loci for free and reduced arrangements.

Provided combinatorial formulas for roots in the tame case.

Linked roots of Bernstein--Sato polynomial to minimal hyperplane additions for freeness.

Abstract

For a central, not necessarily reduced, hyperplane arrangement $f$ equipped with any factorization $f = f_{1} \cdots f_{r}$ and for $f^{\prime}$ dividing $f$, we consider a more general type of Bernstein--Sato ideal consisting of the polynomials $B(S) \in \mathbb{C}[s_{1}, \dots, s_{r}]$ satisfying the functional equation $B(S) f^{\prime} f_{1}^{s_{1}} \cdots f_{r}^{s_{r}} \in \text{A}_{n}(\mathbb{C})[s_{1}, \dots, s_{r}] f_{1}^{s_{1} + 1} \cdots f_{r}^{s_{r} + 1}.$ Generalizing techniques due to Maisonobe, we compute the zero locus of the standard Bernstein--Sato ideal in the sense of Budur (i.e. $f^{\prime} = 1)$ for any factorization of a free and reduced $f$ and for certain factorizations of a non-reduced $f$. We also compute the roots of the Bernstein--Sato polynomial for any power of a free and reduced arrangement. If $f$ is tame, we give a combinatorial formula for the roots lying in $[-1,0).$ For $f^{\prime} \neq 1$ and any factorization of a line arrangement, we compute the zero locus of this ideal. For free and reduced arrangements of larger rank, we compute the zero locus provided $\text{deg}(f^{\prime}) \leq 4$ and give good estimates otherwise. Along the way we generalize a duality formula for $\mathscr{D}_{X,\mathfrak{x}}[S]f^{\prime}f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$ that was first proved by Narv\'aez-Macarro for $f$ reduced, $f^{\prime} = 1$, and $r = 1.$ As an application, we investigate the minimum number of hyperplanes one must add to a tame $f$ so that the resulting arrangement is free. This notion of freeing a divisor has been explicitly studied by Mond and Schulze, albeit not for hyperplane arrangements. We show that small roots of the Bernstein--Sato polynomial of $f$ can force lower bounds for this number.