Generating sequences and Poincaré series for a finite set of plane divisorial valuations
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Generating sequences and Poincaré series for a finite set of plane divisorial valuationsData de publicació
2008Editor
ElsevierISSN
00018708Tipus de document
info:eu-repo/semantics/articleVersió
info:eu-repo/semantics/sumittedVersionParaules clau / Matèries
Resum
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, SV , and the multi-index graded
algebra ... [+]
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, SV , and the multi-index graded
algebra defined by V , grV
R. We prove that SV 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
grV
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 Poincar´e series of V .
Moreover, the Poincar´e series of C could be seen as the limit of the
series of V (k), k ≥ 0 [-]
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