The intrinsic topological nature of the Poincaré series of a plane curve singularity

2023/02/23 by Almirón, Patricio, Moyano-Fernández, Julio-José
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.12079

Abstract

In this paper we provide some factorization theorems of the Poincaré series P_C of a plane curve singularity C depending on some key values of the semigroup of values of \(C\). These results yield an iterative computation of P_C in purely algebraic terms from the dual resolution graph of C. On the other hand, Campillo, Delgado and Gusein-Zade showed in 2003 the equality between P_C and the Alexander polynomial Δ_L of the corresponding link L. Our procedure supplies a new proof of this coincidence. More concretely, we show that our algebraic construction can be translated to the iterated toric structure of the link L. Additionally we show that the semigroup algebra can be defined from the fundamental group of the link exterior in the irreducible case. This gives in particular a conceptual reason for the coincidence of P_C and Δ_L.

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