Mather Discrepancy as an Embedding Dimension in the Space of Arcs

Abstract

Let X be a variety over a field k and let X∞ be its space of arcs. We study the embedding dimension of the completion A^ of the local ring of X∞ at P where P is the stable point defined by a divisorial valuation ν on X. Assuming char k = 0, we prove that the embedding dimension of A^ is equal to k + 1 where k is the Mather discrepancy of X with respect to ν. We also obtain that the dimension of A^ has as lower bound the Mather-Jacobian log-discrepancy of X with respect to ν. For X normal and complete intersection, we prove as a consequence that points P of codimension one in X ∞ have discrepancy k ≤ 0.