Roots of Bernstein–Sato polynomials of certain homogeneous polynomials with two-dimensional singular loci

For homogeneous polynomials of $n$ variables, we present a new method to compute the roots of Bernstein–Sato polynomial supported at the origin, assuming that general hyperplane sections of the associated projective hypersurface have at most weighted homogeneous isolated singularities. Calculating the dimensions of certain $E_r$-terms of the pole order spectral sequence for a given integer $r \in [2, n]$, we can detect its degeneration at $E_r$ for certain degrees. In the case of strongly free, locally positively weighted homogeneous divisors on $\mathbb{P}^3$, we can prove its degeneration almost at $E_2$ and completely at $E_3$ together with a symmetry of a modified pole-order spectrum for the $E_2$-term. These can be used to determine the roots of Bernstein–Sato polynomials supported at the origin, except for rather special cases.

2010 Mathematics Subject Classification

14F10

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This work is partially supported by JSPS Kakenhi 15K04816.

Received 16 March 2017

Accepted 1 March 2019

Published 13 November 2020