On modular inequalities for plane projective curves

Abstract

We introduce modular inequalities for complements of plane curves, based on a Combinatorial Aomoto complex construction associated with the weak combinatorial type of a curve. We use this as a tool to investigate twisted Alexander polynomials, in particular to study the characteristic polynomial of the Milnor fiber associated to a projective plane curve, i.e. the classical Alexander polynomial. We give criteria for the non-triviality of the resonance in degree $1$, over a positive characteristic field, for curves with quasi fiber--type structure and we compute lower bounds for the multiplicities of some roots of twisted Alexander polynomials for this type of curves. We apply these results to theoretically compute the multiplicities of certain roots of twisted Alexander polynomials for fiber--type curves.