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 = f1 ·s fr and for f dividing f, we consider a more general type of Bernstein--Sato ideal consisting of the polynomials B(S) ∈ C[s1, …, sr] satisfying the functional equation B(S) f f1s1 ·s frsr ∈ An(C)[s1, …, sr] f1s1 + 1 ·s frsr + 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 = 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 ≠ 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 deg(f) ≤ 4 and give good estimates otherwise. Along the way we generalize a duality formula for DX,x[S]ff1s1 ·s frsr that was first proved by Narv\'aez-Macarro for f reduced, f = 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.