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- W2950150611 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 Poincar'e 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 Poincar'e series of $V$. Moreover, the Poincar'e series of $C$ could be seen as the limit of the series of $V^{(k)}$, $kge 0$." @default.
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- W2950150611 date "2008-06-17" @default.
- W2950150611 modified "2023-09-27" @default.
- W2950150611 title "Generating sequences and Poincar'e series for a finite set of plane divisorial valuations" @default.
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