Combinatorially determined zeroes of Bernstein--Sato ideals for tame and free arrangements

Daniel Bath

Journal of Singularities
volume 20 (2020), 165-204

Received: 19 October 2019. Received in revised form: 30 May 2020.

DOI: 10.5427/jsing.2020.20h

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Abstract:

For a central, not necessarily reduced, hyperplane arrangement f equipped with any factorization f = f_1 ... f_r and for f' dividing f, we consider a more general type of Bernstein--Sato ideal consisting of the polynomials B(S) \in C[s_1, ..., s_r] satisfying the functional equation B(S)f' f_1^{s_1}...f_r^{s_r} \in A_n(C)[s_1,..., s_r] f_1^{s_1 + 1}...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' = 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 D_{X, x}[S]f'f_1^{s_1}...f_r^{s_r} that was first proved by Narváez-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.

2010 Mathematical Subject Classification:

Primary: 14F10; Secondary: 32S40, 32S05, 32S22, 32C38

Key words and phrases:

Bernstein--Sato, b-function, hyperplane, arrangement, D-module, tame, free divisors, logarithmic, differential, annihilator, Spencer

Author(s) information:

Daniel Bath
Department of Mathematics
Purdue University
West Lafayette, IN, USA
email: dbath@purdue.edu