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

The formal definition of valuation was firstly given by the Hungarian mathematician J. Kürschák in 1912 supported with ideas of Hensel. Valuation theory, based on this concept, has been developed by a large number of contributors (some of them distinguished mathematicians as Krull or Zariski) and it has a wide range of applications in different context and research areas as, for instance, algebraic number theory or commutative algebra and its application to algebraic geometry or theory of diophantine equations. In this paper, we are interested in some applications of valuation theory to algebraic geometry and, particularly, to singularity theory. Valuation theory was one of the main tools used by Zariski when he attempted to give a proof of resolution of singularities for algebraic schemes. In characteristic zero, resolution was proved by Hironaka without using that tool; however there is no general proof for positive characteristic and valuations seem to be suitable algebraic objects for this purpose. Valuations associated with irreducible curve singularities are one of the best known classes of valuations, especially the case corresponding to plane branches where valuations and desingularization process are very related. Germs of plane curves can contain several branches and, for this reason, it is useful to study their corresponding valuations, not only in an independent manner but as a whole [6, 7, 8]. Valuations of the fraction field of some 2-dimensional local regular Noetherian ring R centered at R, that we call plane valuations, are a very interesting class of valuations which includes the above mentioned family related with branches. These valuations were studied by Zariski and their study was revitalized by the paper [46]. Very little is known about valuations in higher dimension. 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, we describe value semigroup, graded algebra and Poincaré 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.