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

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 $f^{\prime }\ne 1$ 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 $\text{deg}(f^{\prime })\le 4$ and give good estimates otherwise. Along the way we generalize a duality formula for ${\mathcal{D}}_{X,\mathfrak{x}}[S]f^{\prime }{f}_1^{s_1}\cdots f_r^{s_r}$ that was first proved by Narváez-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.

arXiv commentRevised for clarity throughout; in particular the abstract and introduction are largely rewritten and some proofs expanded. Note that the last few propositions of Section 2 were removed. (These are true but have more involved proofs than originally presented.) Proposition 2.26 suffices for our purposes. No major results or arguments changed. Final version to appear in Journal of Singularities