# Mather Discrepancy as an Embedding Dimension in the Space of Arcs

### Hussein Mourtada

Institut Mathématique de Jussieu-Paris Rive Gauche, France### Ana J. Reguera

Universidad de Valladolid, Spain

## 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 $k_E \leq 0$.

## Cite this article

Hussein Mourtada, Ana J. Reguera, Mather Discrepancy as an Embedding Dimension in the Space of Arcs. Publ. Res. Inst. Math. Sci. 54 (2018), no. 1, pp. 105–139

DOI 10.4171/PRIMS/54-1-4