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 P E be the stable point of X ∞ defined by a divisorial valuation ν E on X . Assuming char k = 0 , if X is smooth at the center of P E , we make a study of the graded algebra associated to ν E 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 O X ∞ , P E . If X is singular, we obtain generators of P E / P E 2 and conclude that embdim O ( X ∞ ) red , P E = embdim O X ∞ , P E ˆ ≤ k ˆ E + 1 where k ˆ E is the Mather discrepancy of X with respect to ν E . This provides algebraic tools for explicit computations of the local rings O X ∞ , P E ˆ . [ABSTRACT FROM AUTHOR]