Coordinates at stable points of the space of arcs

Let X be a variety over a field k and let X∞ be its space of arcs. Let PE be the stable point of X∞ defined by a divisorial valuation νE on X. Assuming char k=0, if X is smooth at the center of PE, we make a study of the graded algebra associated to νE and define a finite set whose elements generate a localization of the graded algebra modulo étale covering. This provides an explicit description of a minimal system of generators of the local ring OX∞,PE. If X is singular, we obtain generators of PE/PE2 and conclude that embdimO(X∞)red,PE=embdimOX∞,PEˆ≤kˆE+1 where kˆE is the Mather discrepancy of X with respect to νE. This provides algebraic tools for explicit computations of the local rings OX∞,PEˆ.