# Computing Milnor fiber monodromy for some projective hypersurfaces

## TL;DR

This paper introduces an algorithm to compute monodromy and pole order filtration on Milnor fiber cohomology for certain hypersurfaces in projective space, with improved efficiency under specific conditions and supported by theoretical and computational evidence.

## Contribution

The paper presents a new algorithm for computing monodromy and pole order filtration on Milnor fiber cohomology, especially efficient for hyperplane arrangements and free hypersurfaces, supported by conjectures and examples.

## Key findings

- Algorithm computes monodromy and pole order filtration for hypersurfaces.
- Enhanced efficiency for hyperplane arrangements and free hypersurfaces under a conjecture.
- Symmetry observed in pole order spectra for reflection group arrangements.

## Abstract

We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in $\mathbb{P}^n$ whose pole order spectral sequence degenerates at the second page. In the case of hyperplane arrangements and free, locally quasi-homogeneous hypersurfaces, and assuming a key conjecture, this algorithm is much faster than for a hypersurface as above. Our conjecture is supported by the results due to L. Narv\' ez Macarro and M. Saito on the roots of Bernstein-Sato polynomials of such hypersurfaces, by all the examples computed so far, and by one partial result. For hyperplane arrangements coming from reflection groups, a surprising symmetry of their pole order spectra on top cohomology is displayed in our examples. We also improve our previous results in the case of plane curves.