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

Abstract

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

5 pages, LaTeX