Semigroup and Poincar, series for a finite set of divisorial valuations

Let be a finite set of divisorial valuations coming from a modification of , where is a field. In this paper we prove that, if is infinite, its semigroup of values is finitely generated whenever there is some finite generating sequence. In that case, we also show that the Poincar, series associated to is a rational function whose denominator can be expressed in terms of the valuation vectors of the elements in the generating sequence. For this we do not need any assumptions on . However, a finite generating sequence does not always exist. We give an example of a modification whose semigroup of values is not finitely generated.