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

2000/02/07 by Antonio Campillo, A. Campillo, F. Delgado +5
Mathematics · #14H20 #32S05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Geometric Topology (math.GT) #math.AG #math.GT #msc:14H20 #msc:32S05

paper · pdf · doi:10.48550/arxiv.math/0002052

5 pages, LaTeX

arxiv created 2000/02/07 · openalex publication_date 2000/02/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Related

• The Alexander polynomial of a plane curve singularity via the ring of functions on it
• Integration with respect to Euler characteristic over the projectivization of the space of functions and the Alexander polynomial of a plane curve singularity
• Integrals with respect to the Euler characteristic over spaces of functions and the Alexander polynomial
• The Poincare series of divisorial valuations in the plane defines the topology of the set of divisors
• Multivariable Hodge theoretical invariants of germs of plane curves