Roots of Bernstein-Sato polynomials for projective hypersurfaces with general hyperplane sections having weighted homogeneous isolated singularities

For homogeneous polynomials of $n$ variables with singular locus dimension at most 2, we give a new method to compute the roots of Bernstein-Sato polynomials supported at the origin if certain conditions are satisfied. We assume that general hyperplane sections of the associated projective hypersurfaces have at most weighted homogeneous isolated singularities. (This holds trivially if $n=3$.) Calculating the dimensions of certain $E_r$-terms of the pole order spectral sequence, we can detect its partial degeneration at $E_r$ if some relations hold among the dimensions of $E_r$-terms, where $r\in[2,n]$. For the moment there are no examples with last condition unsatisfied when $r=n$, especially for $n=3$, although this may depend on each example if $r=2$. (We might get a partial degeneration at $E_2$ only after calculating certain $E_3$-terms.) This partial degeneration can be used to compute the roots of Bernstein-Sato polynomial supported at the origin if another condition is satisfied. The latter is needed to avoid a difficulty coming from a hypothesis in a theorem we have to use. It holds in the 3 variable case except for polynomials of rather special types as far as calculated.