Generalized Poincaré series for plane curve singularities

Abstract

The purpose of these notes is to give an introduction to the notion of Poincaré series attached to multi-index filtrations associated with (plane) complex curve singularities. This was introduced by Campillo, Delgado and Kiyek in 1994 in a purely algebraic-combinatorial viewpoint and further developed in a series of papers by Campillo, Delgado and Gusein-Zade from 1999 on; in particular, they found an alternative definition as an integral with respect to the Euler characteristic, which allowed them to generalize the concept to a setting of motivic nature in 2011. In this review we will summarize the basic facts and results mainly about the generalized Poincaré series. No claim of novelty —beyond the exposition self— is done.