Given a noetherian local integral domain $R$ and a valuation $\nu$ of its field of fractions which is non negative on $R$, I study a geometric specialization of $R$ to the graded ring ${\rm gr}_\nu R$ determined by the valuation. I show that if the residue field of $R$ is algebraically closed, for 0-dimensional valuations this graded ring corresponds to an essentially toric variety, possibly of infinite embedding dimension, and consider some possible applications of this fact to local uniformization by deformation of a resolution of singularities of the toric variety.

This leads to the study of the $\nu$-adic completion of a noetherian local domain and of the structure and regularity of certain semigroup algebras and graded algebras associated to valuations. A result on the structure of $\nu$-adically complete equicharacteristic noetherian local rings and some phenomena of algebraic geometry in finite Krull dimension but countable embedding dimension play an important role.

Last update: February 9, 1999