Generating sequences and Poincaré series for a finite set of plane divisorial valuations

Abstract

Let V be a finite set of divisorial valuations centered at a 2-dimensional regular local ring R. In this paper we study its structure by means of the semigroup of values, S V , and the multi-index graded algebra defined by V, gr V R . We prove that S V is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in V, the approximation of a reduced plane curve singularity C by families of sets V ( k ) of divisorial valuations, and the relationship between the value semigroup of C and the semigroups of the sets V ( k ) , allow us to obtain the (finite) minimal generating sequences for C as well as for V. We also analyze the structure of the homogeneous components of gr V R . The study of their dimensions allows us to relate the Poincaré series for V and for a general curve C of V. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincaré series of V. Moreover, the Poincaré series of C could be seen as the limit of the series of V ( k ) , k ⩾ 0 .