The Alexander polynomial of a plane curve singularity via the ring of functions on it

Abstract

We prove two formulae that express the Alexander polynomial $\Delta\sp C$ of several variables of a plane curve singularity $C$ in terms of the ring $\mathscr {O}\sb C$ of germs of analytic functions on the curve. One of them expresses $\Delta\sp C$ in terms of dimensions of some factors corresponding to a (multi-indexed) filtration on the ring $\mathscr {O}\sb C$. The other one gives the coefficients of the Alexander polynomial $\Delta\sp C$ as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes).