Subspace Arrangements of Curve Singularities and q-Analogs of the Alexander Polynomial

This paper introduces q-series for a curve singularity (C, 0) in the affine space (Cn, 0) via subspace arrangements. These q-series are certain multivariable generating functions whose coefficients are the characteristic polynomials of subspace arrangements associated with the singularity (C, 0) at various orders; the q is the variable in the characteristic polynomials of the subspace arrangements, representing the equivalence class of the ground field C as an affine algebraic line. We show that all the q-series introduced are rational functions and satisfy some interesting properties. When (C, 0) is a plane curve singularity, the intersection S3 ε ∩ C defines a link Lr, where S3 ε is a 3-sphere centered at the origin with small enough radius ε. The value of certain q-series at q = 1 is the Alexander polynomial of the link Lr, up to a normalization. The approach is to introduce integration on the space OCn,0 of germs via the parameterization of the singularity (C, 0). The whole exposition may be extended to some higher-dimensional singularities.