Poincaré series for curve singularities and its behaviour under projections

Let O be an equicharacteristic reduced complete noetherian local ring of Krull dimension one, and let S be the value semigroup associated with O. The aim of the paper is to investigate the behaviour of the multi-variable Poincaré series associated to S with respect to the property of “forgetting variables”. We prove that, for O Gorenstein, the Poincaré series with one less variable can be explicitly computed in terms of the original series; this provides also a shorter and pure arithmetical way to show that the Poincaré series is a complete invariant of the equisingularity. Moreover we express (without the Gorenstein assumption) the Hilbert series of S in terms of the Poincaré series of the unions of irreducible components of the singularity.The author was partially supported by the German Research Council (DFG), grant GRK 1916 and the Spanish Government Ministerio de Economía y Competitividad (MINECO), grant MTM2012-36917-C03-03