Let \((R, \mathfrak m)\) be a 2-dimensional regular local ring with an algebraically closed coefficient field. A divisorial valuation \(\nu\) is a discrete valuation of the fraction field of \(R\), centered at \((R, \mathfrak m)\). In fact there is a 1-1 correspondance between divisorial valuations and finite sequences of blowing-ups. Let \(V\) be a finite set \(\{\nu_1,\dots,\nu_r\}\) of divisorial valuations centered at a 2-dimensional regular local ring \(R\).
In this paper the authors study its structure by means of the semigroup of values, \(S_V=\{\nu_1(f),\dots,\nu_r(f), f\in R\setminus \{0\}\}\subset \mathbb Z^r\), and the multi-index graded algebra defined by \(V\), gr\(_V\) \(R\). They prove that \(S_V\) is finitely generated and they compute its minimal set of generators following the study of reduced curve singularities. Moreover, they 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 the authors to obtain the (finite) minimal generating sequences for \(C\) as well as for \(V\).
They also analyze the structure of the homogeneous components of gr\(_V\) \(R\). The study of their dimensions allows the authors 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, they 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 \(V^{(k)}, k\geqslant 0\).