Mather Discrepancy as an Embedding Dimension in the Space of Arcs

Abstract

Let $X$ be a variety over a field $k$ and let $X_\infty$ be its space of arcs. We study the complete local ring $\widehat{A}:=\widehat{{\cal O}_{X_\infty, P_{eE}}}$, where $P_{eE}$ is the stable point defined by an integer $e \geq 1$ and a divisorial valuation $\nu_E$ on $X$. Assuming char $k =0$, we prove that embdim $\widehat{A} = e ( \widehat{k}_E + 1)$, where $\widehat{k}_E$ is the Mather discrepancy of $X$ with respect to $\nu_E$. We also obtain that dim $\widehat{A}$ has as lower bound $e ( a_{\rm {MJ}}(E;X))$, where $a_{\rm {MJ}}(E;X)$ is the Mather–Jacobian log-discrepancy of $X$ with respect to $\nu_E$. For $X$ normal and a complete intersection, we prove as a consequence that if $P_E$ has codimension 1 in $X_\infty$ then the discrepancy $k_E \leq 0$.