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 $\hat{A}:=\widehat{{\mathcal{O}}_{X_{\infty },P_{eE}}}$, where $P_{eE}$ is the stable point defined by an integer $e\ge 1$ and a divisorial valuation ${\nu }_E$ on $X$. Assuming char $k=0$, we prove that embdim $\hat{A}=e({\hat{k}}_E+1)$, where ${\hat{k}}_E$ is the Mather discrepancy of $X$ with respect to ${\nu }_E$. We also obtain that dim $\hat{A}$ has as lower bound $e(a_{\mathrm{MJ}}(E;X))$, where $a_{\mathrm{MJ}}(E;X)$ 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 $k_E\le 0$.