Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves

TL;DR

This paper presents an algorithm to compute the monodromy and pole order filtration on the Milnor fiber cohomology of plane curves, linking it to Bernstein-Sato polynomials, with results supported by computational examples.

Contribution

It introduces a novel algorithm for calculating monodromy and pole order filtration on Milnor fiber cohomology of plane curves, including cases with non weighted homogeneous singularities.

Findings

Algorithm successfully computes filtrations for various plane curves.

Results support the conjecture in all computed examples.

Links between Milnor fiber cohomology and Bernstein-Sato polynomial are established.

Abstract

We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve $C$. The relation to the zero set of Bernstein-Sato polynomial of the defining homogeneous polynomial for $C$ is also discussed. When $C$ has some non weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds.