Homogeneous singularity and the Alexander polynomial of a projective plane curve

Abstract

The Alexander polynomial of a plane curve is an important invariant in global theories on curves. However, it seems that this invariant and even a much stronger one the fundamental group of the complement of a plane curve may not distinguish non-reduced curves. In this article, we consider a general problem which concerns a hypersurface of the complex projective space $\mathbb P^n$ defined by an arbitrary homogeneous polynomial $f$. The singularity of $f$ at the origin of $\mathbb C^{n+1}$ is studied, by means of the characteristic polynomials $\Delta_l(t)$ of the monodromy, and via the relation between the monodromy zeta function and the Hodge spectrum. Especially, we go further with $\Delta_1(t)$ in the case $n=2$ and aim to regard it as an alternative object of the Alexander polynomial for $f$ non-reduced. This work is based on knowledge of multiplier ideals and local systems.