Publications of the Research Institute for Mathematical Sciences

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Volume 54, Issue 1, 2018, pp. 105–139
DOI: 10.4171/PRIMS/54-1-4

Published online: 2018-01-17

Mather Discrepancy as an Embedding Dimension in the Space of Arcs

Hussein Mourtada[1] and Ana J. Reguera[2]

(1) Institut Mathématique de Jussieu-Paris Rive Gauche, France
(2) Universidad de Valladolid, Spain

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 $k_E \leq 0$.

Keywords: Space of arcs, divisorial valuations, embedding dimension, Mather discrepancy

Mourtada Hussein, Reguera Ana: Mather Discrepancy as an Embedding Dimension in the Space of Arcs. Publ. Res. Inst. Math. Sci. 54 (2018), 105-139. doi: 10.4171/PRIMS/54-1-4