Finite families of plane valuations: value semigroup, graded algebra and Poincaré series

In this paper, the authors are interested in some applications of valuation theory to algebraic geometry and, particularly, to singularity theory. The aim of this paper is to provide a concise survey of some aspects of the theory of plane valuations, adding some comments upon more general valuations when it is possible. For those valuations, authors describe value semigroup, graded algebra and Poincar ́e series emphasizing on the recent study of the same algebraic objects for finite families of valuations and their relation with the corresponding ones for reduced germs of plane curves